Answer:
The total cost of resurface the path is [tex]\$600[/tex]
Step-by-step explanation:
step 1
Find the area of the path
The area of the path is equal to the area of the path plus the swimming pool minus the area of the swimming pool
[tex]A=(14+3+3)(30+3+3)-(14)(30)[/tex]
[tex]A=(20)(36)-(14)(30)[/tex]
[tex]A=300\ ft^{2}[/tex]
step 2
Find the cost of resurface the path
Multiply the area of the path by $2 per square foot
[tex]300*2=\$600[/tex]
the total cost of resurfacing the path is $600.
To calculate the total cost of resurfacing the path around the swimming pool, you first need to determine the area of the path. The swimming pool measures 14 feet by 30 feet, and the path is 3 feet wide. To find the area of the outer rectangle, which includes the pool and the path, you calculate the width and length including the path. This gives you a width of (14 + 2*3) feet and a length of (30 + 2*3) feet, as the path goes all the way around, adding twice the width of the path to each dimension.
The outer rectangle's dimensions are therefore 20 feet by 36 feet. The area of the outer rectangle is 20 feet * 36 feet = 720 square feet. The area of the pool itself is 14 feet * 30 feet = 420 square feet. To find the area of just the path, you subtract the area of the pool from the area of the outer rectangle: 720 square feet - 420 square feet = 300 square feet. The cost to resurface the path is $2 per square foot, so the total cost is 300 square feet * $2/square foot = $600.
A proton moves at 5.20 105 m/s in the horizontal direction. It enters a uniform vertical electric field with a magnitude of 8.40 103 N/C. Ignore any gravitational effects. (a) Find the time interval required for the proton to travel 4.50 cm horizontally.
Answer:
86.5 ns
Step-by-step explanation:
The speed in the original direction (horizontally) is unchanged by the vertical force the field exerts. The travel time is ...
time = distance/speed = (4.5×10^-2 m)/(5.20×10^5 m/s) = 8.65×10^-8 s
_____
An engineer would express this time using the SI prefix nano- for 10^-9. The time is 86.5 ns.
Desmond wants to sell his car that he paid $8,000 for 2 years ago. The car depreciated, or decreased in value, at a constant rate each month over a 2-year period. If x represents the monthly depreciation amount, which expression shows how much Desmond can sell his car for today? 8,000 + 24x 8,000 - 24x 8,000 + 2x 8,000 - 2xDesmond wants to sell his car that he paid $8,000 for 2 years ago. The car depreciated, or decreased in value, at a constant rate each month over a 2-year period. If x represents the monthly depreciation amount, which expression shows how much Desmond can sell his car for today? 8,000 + 24x 8,000 - 24x 8,000 + 2x 8,000 - 2x
Answer:
8,000 - 24x
Step-by-step explanation:
Let
y ----> depreciated value of the car
x---> rate of depreciation
t ----> the time in months
we know that
The linear equation that represent this situation is
y=8,000-xt
For t=2 years=2*12=24 months
substitute
y=8,,000-x(24)
y=8,000-24x
Answer:
Step-by-step explanation:
The answer is B) 8,000-24xThe indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5) in Section 4.2, y2 = y1(x) e−∫P(x) dx y 2 1 (x) dx (5) as instructed, to find a second solution y2(x). y'' + 2y' + y = 0; y1 = xe−x
First confirm that [tex]y_1=xe^{-x}[/tex] is a solution to the ODE,
[tex]y''+2y'+y=0[/tex]
We have
[tex]{y_1}'=e^{-x}-xe^{-x}=(1-x)e^{-x}[/tex]
[tex]{y_1}''=-e^{-x}-(1-x)e^{-x}=(-2+x)e^{-x}[/tex]
Substituting into the ODE gives
[tex](-2+x)e^{-x}+2(1-x)e^{-x}+xe^{-x}=0[/tex]
Suppose [tex]y_2(x)=v(x)y_1(x)[/tex] is another solution to this ODE. Then
[tex]{y_2}'=v'y_1+v{y_1}'[/tex]
[tex]{y_2}''=v''y_1+2v'{y_1}'+v{y_1}''[/tex]
and substituting these into the ODE yields
[tex](v''y_1+2v'{y_1}'+v{y_1}'')+2(v'y_1+v{y_1}')+vy_1=0[/tex]
[tex]xe^{-x}v''+2e^{-x}v'=0[/tex]
[tex]xv''+2v'=0[/tex]
Let [tex]w(x)=v'(x)[/tex]. Then the remaining ODE is linear in [tex]w[/tex]:
[tex]xw'+2w=0[/tex]
Multiply both sides by the integrating factor, [tex]x[/tex], and condense the left hand side as a derivative of a product:
[tex]x^2w'+2xw=(x^2w)'=0[/tex]
Integrate both sides with respect to [tex]x[/tex] and solve for [tex]w[/tex]:
[tex]x^2w=C_1\implies w=C_1x^{-2}[/tex]
Back-substitute and integrate both sides with respect to [tex]x[/tex] to solve for [tex]v[/tex]:
[tex]v'=C_1x^{-2}\implies v=-C_1x^{-1}+C_2[/tex]
Back-substitute again to solve for [tex]y_2[/tex]:
[tex]\dfrac{y_2}{y_1}=C_2-\dfrac{C_1}x[/tex]
[tex]\implies y_2=C_2xe^{-x}-C_1e^{-x}[/tex]
[tex]y_1[/tex] already captures the solution [tex]xe^{-x}[/tex], so the remaining one is
[tex]\boxed{y_2=e^{-x}}[/tex]
A differential equation shows the relationship between functions and their derivatives.
The equation of [tex]y_2(x)[/tex] is: [tex]y_2 = -e^{-x}[/tex]
The given parameters are:
[tex]y_2 = y_1(x) \int\frac{ e^{(\int -P(x)\ dx)} }{ y_1^2(x) }dx[/tex]
[tex]y" + 2y' + y = 0[/tex]
[tex]y_1 = xe^{-x}[/tex]
The general equation is:
[tex]y" + P(x) y' + Q(x)y = 0[/tex]
Compare the above equation to [tex]y" + 2y' + y = 0[/tex]
[tex]P(x) = 2[/tex]
Integrate:
[tex]\int\limits^x_0 P(x') dx'= \int\limits^x_0 2 dx'[/tex]
[tex]\int\limits^x_0 P(x') dx'= 2x|\limits^x_0[/tex]
[tex]\int\limits^x_0 P(x') dx'= 2[x - 0][/tex]
[tex]\int\limits^x_0 P(x') dx'= 2[x ][/tex]
[tex]\int\limits^x_0 P(x') dx'= 2x[/tex]
We have:
[tex]y_2 = y_1(x) \int\frac{ e^{(\int -P(x)\ dx)} }{ y_1^2(x) }dx[/tex]
The above equation becomes:
[tex]y_2 = y_1(x) \int\frac{e^{(\int -2 dx)} }{ y_1^2(x) }dx[/tex]
Substitute [tex]y_1 = xe^{-x}[/tex]
[tex]y_2 = xe^{-x} \int\frac{ e^{(\int -2 dx)} }{ (xe^{-x})^2 }dx[/tex]
Integrate
[tex]y_2 = xe^{-x} \int\frac{ e^{-2x}}{ (xe^{-x})^2 }dx[/tex]
Evaluate the exponents
[tex]y_2 = xe^{-x} \int\frac{ e^{-2x}}{ x^2e^{-2x} }dx[/tex]
Cancel out common factors
[tex]y_2 = xe^{-x} \int\frac{1}{ x^2 }dx[/tex]
Rewrite as:
[tex]y_2 = xe^{-x} \int x^{-2}dx[/tex]
Integrate
[tex]y_2 = xe^{-x} \times -\frac{1}{x} + c[/tex]
[tex]y_2 = -e^{-x} + c[/tex]
Set c to 0.
[tex]y_2 = -e^{-x} + 0[/tex]
[tex]y_2 = -e^{-x}[/tex]
Hence, the equation of [tex]y_2(x)[/tex] is:
[tex]y_2 = -e^{-x}[/tex]
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is an = -n + 2 a solution to an = an-1 +2an-2 + 2n-9
Answer:
Yes, [tex]a_n=-n+2[/tex] is a solution of [tex]a_n=a_{n-1}+2a_{n-2}+2n-9[/tex].
Step-by-step explanation:
Given
[tex]a_n= a_{n-1}+2a_{n-2}+2n-9[/tex]
To prove that [tex]a_n=-n+2[/tex] is a solution of the given expression [tex]a_n=a_{n-1}+2a_{n-2}+2n-9[/tex]
Take Right hand side :
[tex]a-{n-1}+2a_{n-2}+2n-9[/tex]
Substitute the value of [tex]a_n=-n+2 [/tex]
Now , we get
-(n-1)+2+2{-(n-2)+2}+2n-9
=-n+1+2+2(-n+4)+2n-9 ( simplified)
=-n-6+8-2n+2n ( simplified )
=[tex]-n+2 ( simplified)=a_n[/tex]
Hence, LHS=RHS
Therefore, [tex]a_n=-n+2[/tex] is a solution of given expression
[tex]a_n=a_{n-1}+2a_{n-2}+2n-9[/tex]
Yes, [tex]a_n=-n+2 [/tex] is a solution of given expression.
Upon substituting and simplifying the given expressions, it's confirmed that an = -n + 2 is indeed a solution to the equation an = an-1 + 2an-2 + 2n - 9.
Explanation:The question asks if an = -n + 2 is a solution to an = an-1 + 2an-2 + 2n - 9. To verify, let's substitute an into the right side of the equation.
Substitute an-1 = -(n-1) + 2 and an-2 = -(n-2) + 2 into the equation:
an = (-(n-1) + 2) + 2(-(n-2) + 2) + 2n - 9.
Simplifying, an = -n + 2. Therefore, an = -n + 2 satisfies the given equation, confirming it as a solution.
Question 7 4 pts To borrow money, you pawn your mountain bike. Based on the value of the bike, the pawnbroker loans you $552. One month later, you get the bike back by paying the pawnbroker $851. What annual interest rate did you pay? Write as a percentage.
Answer:
The annual rate of interest is 650 %.
Step-by-step explanation:
Given,
The amount of loan = $ 552,
Total amount paid after one month = $ 851,
So, the interest for one month = $ 851 -$ 552 = $ 299,
Thus, the monthly interest = [tex]\frac{\text{Interest for a month}}{\text{Total amount of loan}}\times 100[/tex]
[tex]=\frac{299}{552}\times 100[/tex]
[tex]=\frac{29900}{552}[/tex]
Since, 1 year = 12 month ⇒ 1 month = 1/12 year,
Hence, the annual rate of interest = [tex]\frac{29900}{552}\times 12=\frac{358800}{552}=650\%[/tex]
Answer:
650.2%
Step-by-step explanation:
We have to calculate annual interest rate by this formula :
A = P( 1 + rt )
A = Future value of loan ( $851 )
P = Principal amount ( $552 )
r = Rate of interest
t = Time in years
As we know, 1 year = 12 months . By converting 1 month to year we get
1 month = [tex]\frac{1}{12}[/tex] year = 0.0833 year
Now we put the values in the formula
$851 = $552( 1 + r × 0.0833 )
= [tex]\frac{851}{552} =\frac{552(1+(0.0833r))}{552}[/tex]
= 1.5417 = 1 + 0.0833r
= 1.5417 - 1 = 0.0833r
= 0.5417 = 0.0833r
r = 6.502
r is in decimal form so we have to multiply with 100 to convert the value in percentage.
6.502 × 100 = 650.2%
The annual interest rate that you pay is 650.2%
Two cyclists start at the same point and travel in opposite directions. One cyclist travels 10 km/ h slower than the other. If the two cyclists are 120 km apart after 2 hours, what is the rate of each cyclist?
Answer:
A⇒35 km/hr
B⇒25 km/hr
Step-by-step explanation:
Let us call them cyclist A and B.
The rate of cyclist A will be x km/h
The rate of cyclist B will be (x-10) km/h
The total distance traveled= 120 km
Time taken =2 hours
Relative speed, since they are travelling in different directions, is arrived at by addition of their individual speeds = {x+ (x-10)}km/hr (2x-10)km/hr
Distance =speed × time
120km = (2x-10)km/hr×2
120=4x-20
4x=140
x= 35 km/hr⇒ speed for cyclist A
B⇒ (x-10) km/hr=35-10=25 km/hr
The rate of the faster cyclist is 35 km/h and the rate of the slower cyclist is 25 km/h.
Explanation:The context of this problem is based on understanding relative speed, also known as the rate at which two objects move away from each other when traveling in opposite directions. We know that the two cyclists are 120 km apart after 2 hours, meaning together they covered a total distance of 120 km.
Let's denote the speed of the faster cyclist as 'x' km/h and the speed of the slower cyclist as 'x - 10' km/h. As they are moving in opposite directions, their speeds add up, and we can use this to form an equation. For the total distance they covered, we can write it as time multiplied by their combined speed, which is (x + (x - 10)) km/h. Thus:
2 * (x + x - 10) = 120
This simplifies to:
2 * (2x - 10) = 120
When we solve for 'x', we get:
2x - 10 = 60
2x = 70
x = 35
Therefore, the rate of the faster cyclist is 35 km/h and the rate of the slower cyclist is '35 - 10' which is 25 km/h.
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If two polygons are similar with the lengths of their corresponding sides in the ratio of a/b , then the ratio of their areas is _____.
Answer:
The ratio of their areas is [tex]\frac{a^{2}}{b^{2}}[/tex]
Step-by-step explanation:
we know that
If two figures are similar, then the ratio of its corresponding sides is proportional and this ratio is called the scale factor and the ratio of its areas is equal to the scale factor squared
Let
z -----> the scale factor
In this problem the scale factor is equal to
[tex]z=\frac{a}{b}[/tex]
therefore
the scale factor squared is equal to
[tex]z^{2} =\frac{a^{2}}{b^{2}}[/tex]
Answer:
a^2/b^2
Step-by-step explanation:
Answer key for the assignment on the website. (Can’t say it because it’s for some reason censored on here)
. A normal population has a mean of 35 and a standard deviation of 8. a. What proportion of the population is between 20 and 30? b. What is the probability that a randomly chosen value will be between 30 and 40? solve in R
Answer: a. The proportion of the population is between 20 and 30 =0.2356
b. The probability that a randomly chosen value will be between 30 and 40 =0.4680
Step-by-step explanation:
Given : Mean : [tex]\mu=35[/tex]
Standard deviation : [tex]\sigma = 8[/tex]
The formula to calculate z-score :-
[tex]z=\dfrac{x-\mu}{\sigma}[/tex]
For x= 20
[tex]z=\dfrac{20-35}{8}=-1.875[/tex]
For x= 30
[tex]z=\dfrac{30-35}{8}=-0.625[/tex]
For x= 40
[tex]z=\dfrac{40-35}{8}=0.625[/tex]
a.
[tex]P(20<x<30)=P(-1.875<z<-0.625)\\= P(-0.625)-P(-1.875)\\=0.2659855-0.0303964=0.2355891\approx0.2356[/tex]
b.
[tex]P(30<x<40)=P(-0.625<z<0.625)\\= P(0.625)-P(0.625)\\=0.7340144-0.2659855=0.4680289\approx0.4680[/tex]
We can use the pnorm() function in the R programming language to calculate the proportion of a population and the probability of a randomly chosen value in a normal distribution, giving us the proportion between 20 and 30, and the probability between 30 and 40.
Explanation:In the R programming language, we can solve this problem by using the pnorm() function, which represents the cumulative distribution function for a normal distribution. The function pnorm(x, mean, sd) gives the probability that a normally distributed random number will be less than x.
To answer your questions:
a. What proportion of the population is between 20 and 30?
First, we need to calculate the probability (which represents proportion in this case) for 30 and 20 separately. The code in R will be pnorm(30, 35, 8) - pnorm(20, 35, 8). This gives us the proportion of the population that is between 20 and 30.
b. What is the probability that a randomly chosen value will be between 30 and 40?
Similarly, for this we can use pnorm(40, 35, 8) - pnorm(30, 35, 8). This gives us the probability that a randomly chosen value will be between 30 and 40. Learn more about Normal Distribution here:
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The city of Raleigh has 10500 registered voters. There are two candidates for city council in an upcoming election: Brown and Feliz. The day before the election, a telephone poll of 450 randomly selected registered voters was conducted. 237 said they'd vote for Brown, 190 said they'd vote for Feliz, and 23 were undecided.a. What is the population of this survey? b. What is the size of the population?c. What is the size of the sample?e. Give the sample statistic for the proportion of voters surveyed who said they'd vote for Brown. f. Based on this sample, we might expect how many of the 9500 voters to vote for Browng. Is this data qualitative or quantitative?
Answer:
quantitative
Step-by-step explanation:
Calculate the annual effective interest rate of a 12 % nominal annual interest rate compound monthly
Answer:
12.683%
Step-by-step explanation:
The effective annual rate is given by ...
(1 +r/n)^n -1
where r is the nominal annual rate, and n is the number of compoundings per year. Filling in the given numbers, we have ...
effective rate = (1 +0.12/12)^12 -1 ≈ 0.12683 = 12.683%
how to graph each pair of parametric equations using the graphing calculator? x = 3 sin^3t y = 3 cos^3t
The details will depend on your calculator. Attached is a graph using the Desmos calculator.
(08.03 LC) Factor completely: x2 + 10x + 24 (5 points) Prime (x + 12)(x + 2) (x + 3)(x + 8) (x + 6)(x + 4)
Answer:
Option C is correct
Step-by-step explanation:
We need to factorize the expression:
[tex]x^2+10x+24[/tex]
For factorization we need to break the middle term such that the product is equal to 24x^2 and the sum is equal to 10x
We know that 6*4 = 24 and 6+4 =10
So, solving
[tex]=x^2+6x+4x+24[/tex]
Taking common
[tex]=x(x+6)+4(x+6)[/tex]
[tex]=(x+4)(x+6)[/tex]
So, the factors of
[tex]x^2+10x+24[/tex]
are
[tex](x+4)(x+6)[/tex]
Hence Option C is correct
The correct answer is actually D. The person who had answered before me had accurate math, but they were confused. The answer
(x + 6)(x + 4) is on D, not C.
Find the volume of the wedge cut from the first octant by the cylinder z=12-3y^2 and the plane x+y=2.
Answer:
The wedge cut from the first octant ⟹ z ≥ 0 and y ≥ 0 ⟹ 12−3y^2 ≥ 0 ⟹ 0 ≤ y ≤ 2
0 ≤ y ≤ 2 and x = 2-y ⟹ 0 ≤ x ≤ 2
V = ∫∫∫ dzdydx
dz has changed from zero to 12−3y^2
dy has changed from zero to 2-x
dx has changed from zero to 2
V = ∫∫∫ dzdydx = ∫∫ (12−3y^2) dydx = ∫ 12(2-x)-(2-x)^3 dx =
24(2)-6(2)^2+(2-2)^4/4 -(2-0)^4/4 = 20
Step-by-step explanation:
It can be deduced that the volume of the wedge cut from the first octant will be 20.
How to calculate the volumeFrom the information, the wedge cut from the first octant will be z ≥ 0 and y ≥ 0 = 12−3y² ≥ 0 = 0 ≤ y ≤ 2
Also, it can be deduced that 0 ≤ y ≤ 2 and x = 2-y ⟹ 0 ≤ x ≤ 2. Therefore, V = ∫dzdydx. In this case,
dz = 12−3y
dy = 2-x
dx = 2
V = ∫dzdydx = ∫(12−3y²) dydx = ∫12(2-x)-(2-x)³dx
= [24(2) - 6(2)² + (2-2)⁴/4 -(2-0)⁴]/4
= 20
In conclusion, the volume of the wedge cut from the first octant will be 20.
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Oscar has lost his dog in either forest A (with a priori probability 0.4) or in forest B (with a priori probability 0.6). On any given day, if the dog is in A and Oscar spends a day searching for it in A, the conditional probability that he will find the dog that day is 0.25. Similarly, if the dog is in B and Oscar spends a day looking for it there, the conditional probability that he will find the dog that day is 0.15. The dog cannot go from one forest to the other. Oscar can search only in the daytime, and he can travel from one forest to the other only at night. In which forest should Oscar look to maximize the probability he finds his dog on the first day of the search? Given that Oscar looked in A on the first day but didn't find his dog, what is the probability that the dog is in A? If Oscar flips a fair coin to determine where to look on the first day and finds the dog on the first day, what is the probability that he looked in A?
Answer:
a
Step-by-step explanation:
still a 10% chance for oscar
a) Oscar should look in forest A to maximize the probability of finding his dog on the first day. b) So, the probability that the dog is in forest A, given that Oscar looked in A on the first day but didn't find his dog, is [tex]\( \frac{1}{3} \)[/tex]. c) The probability that Oscar looked in forest A, given that he finds the dog on the first day, is approximately [tex]\(0.6579\)[/tex].
a) To maximize the probability of finding his dog on the first day, Oscar should choose the forest where the conditional probability of finding the dog on that day is higher.
For forest A:
Probability of finding the dog on the first day = Probability of the dog being in forest A × Conditional probability of finding the dog in forest A on the first day
[tex]\[ = 0.4 \times 0.25 = 0.1 \][/tex]
For forest B:
Probability of finding the dog on the first day = Probability of the dog being in forest B × Conditional probability of finding the dog in forest B on the first day
[tex]\[ = 0.6 \times 0.15 = 0.09 \][/tex]
Comparing these probabilities, Oscar should look in forest A to maximize the probability of finding his dog on the first day.
b) Given that Oscar looked in forest A on the first day but didn't find his dog, we need to find the conditional probability that the dog is in forest A.
Using Bayes' theorem:
[tex]\[ P(\text{Dog in A} | \text{Search in A, not found}) = \frac{P(\text{Search in A, not found} | \text{Dog in A}) \times P(\text{Dog in A})}{P(\text{Search in A, not found})} \][/tex]
[tex]\[ = \frac{(1 - 0.25) \times 0.4}{1 - (0.4 \times 0.25)} \][/tex]
[tex]\[ = \frac{0.75 \times 0.4}{1 - 0.1} \][/tex]
[tex]\[ = \frac{0.3}{0.9} \][/tex]
[tex]\[ = \frac{1}{3} \][/tex]
c) If Oscar flips a fair coin to determine where to look on the first day and finds the dog on the first day, we need to find the probability that he looked in forest A.
Let [tex]\( H \)[/tex] be the event that the coin flip resulted in heads (i.e., he looked in forest A), and [tex]\( D \)[/tex] be the event that he finds the dog on the first day.
Using Bayes' theorem again:
[tex]\[ P(H | D) = \frac{P(D | H) \times P(H)}{P(D)} \][/tex]
[tex]\[ = \frac{P(D | H) \times P(H)}{P(D | A) \times P(A) + P(D | B) \times P(B)} \][/tex]
Since he finds the dog on the first day, [tex]\( P(D) = P(D | A) \times P(A) + P(D | B) \times P(B) \)[/tex].
[tex]\[ P(D) = (0.25 \times 0.4) + (0.15 \times 0.6) \][/tex]
[tex]\[ P(D) = 0.1 + 0.09 \][/tex]
[tex]\[ P(D) = 0.19 \][/tex]
Now, substitute the values:
[tex]\[ P(H | D) = \frac{0.25 \times 0.5}{0.19} \][/tex]
[tex]\[ P(H | D) = \frac{0.125}{0.19} \][/tex]
[tex]\[ P(H | D) \approx 0.6579 \][/tex]
The complete question is
Oscar has lost his dog in either forest A (with a priori probability 0.4) or in forest B (with a priori probability 0.6). On any given day, if the dog is in A and Oscar spends a day searching for it in A, the conditional probability that he will find the dog that day is 0.25. Similarly, if the dog is in B and Oscar spends a day looking for it there, the conditional probability that he will find the dog that day is 0.15. The dog cannot go from one forest to the other. Oscar can search only in the daytime, and he can travel from one forest to the other only at night.
a) In which forest should Oscar look to maximize the probability he finds his dog on the first day of the search?
b) Given that Oscar looked in A on the first day but didn't find his dog, what is the probability that the dog is in A?
c) If Oscar flips a fair coin to determine where to look on the first day and finds the dog on the first day, what is the probability that he looked in A?
The United States uses 1.4×1019 J of electrical energy per year. Assume, that all this energy came from the fission of 235U, which releases 200 MeV per fission event. Assume that all fission energy is converted into electrical energy. Part A How many kilograms of 235U would be used per year? Express your answer to two significant figures and include the appropriate units. m m = nothing nothing Request Answer Part B How many kilograms of uranium would have to be mined per year to provide that much 235U? (Recall that only 0.70% of naturally occurring uranium is 235U.) Express your answer to two significant figures and include the appropriate units.
Answer:
r55
Step-by-step explanation:
all this energy came from the fission of 235U, which releases 200 MeV per fission event. Assume that all fission energy is converted into electrical energy. Part A How ma
use the second shift theorem to find the inverse laplace transform of (e^−3s) / (s^2 + 2s − 3)
Answer:
see attachment
Step-by-step explanation:
Using the second shift theorem, the inverse Laplace transform of (e^{-3s}) / (s² + 2s - 3) is the convolution of 1 with e^{-4t}, yielding e^{-4t} as the inverse transform.
To find the inverse Laplace transform of (e^{-3s}) / (s² + 2s - 3) using the second shift theorem, we first need to consider the standard form of the second shift theorem: L(e^{at}y(t)) = Y(s-a).
Given that the numerator of our function is e^{-3s}, this implies a shift in the function y(t) by 3 units to the right in the time domain.
Next, we factor the denominator to find its roots: s² + 2s - 3 factors into (s+3)(s-1). Hence, we can rewrite the denominator as a product of first-order terms.
Now, by shifting the transform back by 3 units using the second shift theorem, we obtain the inverse Laplace transform L^{-1}{1/(s+3-(-3))(s-1-(-3))} = L^{-1}{1/s(s+4)}.
This further translates to the convolution of the inverse transforms of 1/s and 1/(s+4), which correspond to the functions 1 and e^{-4t}, respectively, in the time domain.
Thus, the inverse Laplace transform sought is the convolution of 1 with e^{-4t}, which is e^{-4t}.
g Determine the point estimate of the population mean and margin of error for the confidence interval. Lower bound is 21, upper bound is 25. The point estimate of the population mean is nothing. The margin of error for the confidence interval is nothing.
Answer: The point estimate of the population mean is 23. The margin of error for the confidence interval is 2.
Step-by-step explanation:
The confidence interval for the mean is of the form is given by :-
[tex]\overline{x}-\text{error}\leq\mu\leq\overline{x}+\text{error}[/tex]
Given : Lower bound: [tex]\overline{x}-\text{error}=21[/tex]
Upper bound :[tex]\overline{x}+\text{error}= 25[/tex]
Then ,the sum of lower and upper bound will be :-
[tex]2\overline{x}=21+25=46\\\\\Rightarrow\ \overline{x}=23[/tex]
Since [tex]\overline{x}-\text{error}=21[/tex]
[tex]\Rightarrow\ \text{Error}=\overline{x}-21=23-21=2[/tex]
Hence, The point estimate of the population mean is 23. The margin of error for the confidence interval is 2.
The point estimate of the population mean is calculated as the mid-point of the interval, which is 23. The margin of error for the confidence interval is calculated as the distance between the point estimate and either end of the interval, which is 2.
Explanation:To determine the point estimate of the population mean and the margin of error for the confidence interval, we first need to understand the form of a confidence interval. A confidence interval generally has the form (lower bound, upper bound) = (point estimate – margin of error, point estimate + margin of error).
In this case, with a given lower bound as 21 and upper bound as 25, we can perform a simple calculation to find the point estimate which is the mid-point of the interval. It's given by (lower bound + upper bound) / 2, which is (21+25)/2 = 23.
The margin of error for the confidence interval is calculated as the distance between the point estimate and either end of the interval. This is given by (upper bound - point estimate) or (point estimate - lower bound). It is therefore, (25-23) = 2 or (23-21) = 2. So, the point estimate of the population mean is 23 and the margin of error for the confidence interval is 2.
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The sum of the circumference of a circle and the perimeter of a square is 24. Find the dimensions of the circle and square that produce a minimum total area. (Let x be the length of a side of the square and r be the radius of the circle.)
Answer:
The radius of the circle is [tex]r=1.68\ units[/tex]
The length of the square is [tex]x=3.36\ units[/tex]
Step-by-step explanation:
we know that
The circumference of a circle is equal to [tex]C=2\pi r[/tex]
The perimeter of the square is equal to [tex]P=4x[/tex]
so
[tex]24=2\pi r+4x[/tex]
Simplify
[tex]12=\pi r+2x[/tex]
[tex]x=(12-\pi r)/2[/tex] -----> equation A
The area of a circle is equal to [tex]A=\pi r^{2}[/tex]
The area of a square is [tex]A=x^{2}[/tex]
The total area is equal to
[tex]At=\pi r^{2}+x^{2}[/tex] -----> equation B
substitute equation A in equation B
[tex]At=\pi r^{2}+[(12-\pi r)/2]^{2}[/tex]
This is a vertical parabola open upward
The vertex is the minimum
The x-coordinate of the vertex is the radius of the circle that produce a minimum area
The y-coordinate of the vertex is the minimum area
Solve by graphing
The vertex is the point (1.68, 20.164)
see the attached figure
therefore
The radius of the circle is
[tex]r=1.68\ units[/tex]
Find the value of x
[tex]x=(12-\pi r)/2[/tex]
assume
[tex]\pi =3.14[/tex]
[tex]x=(12-(3.14)*(1.68))/2[/tex]
[tex]x=3.36\ units[/tex]
To find the minimum total area, set up equations for the perimeter/circumference and area of the square and circle. Then differentiate the total area and set equal to zero.
Explanation:To find the dimensions of the circle and square that produce a minimum total area, you'll need to use differential calculus. The first thing to know is that the circumference of a circle = 2πr and the perimeter of a square = 4x, where r is the radius of the circle and x is the length of a side of the square. Given that these two add up to 24, we set these equal to each other to get an equation 2πr + 4x = 24.
Next, the total area A of the circle and the square is πr²+ x², and we are asked to minimize this area. So, we need to differentiate the total area A with respect to r and set the result equal to 0 to find the minimal area solution.
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A student is taking two courses, history and math. The probability the student will pass the history course is .54, and the probability of passing the math course is .61. The probability of passing both is .43. What is the probability of passing at least one
The probability of passing at least one is:
0.72
Step-by-step explanation:Let A denote the event that the student passed in history subject.
Let B denote the event that the student passed in math subject.
Then A∪B denote the event that he passed in atleast one.
A∩B denote the event that he passed in both.
We know that:
[tex]P(A\bigcup B)=P(A)+P(B)-P(A\bigcap B)[/tex]
From the ques we have:
[tex]P(A)=0.54\\\\P(B)=0.61\\\\and\\\\P(A\bigcap B)=0.43[/tex]
Hence, we get:
[tex]P(A\bigcup B)=0.54+0.61-0.43\\\\P(A\bigcup B)=0.72[/tex]
Hence, the answer is:
0.72
Consider the following equation. f(x, y) = e−(x − a)2 − (y − b)2 (a) Find the critical points. (x, y) = a,b (b) Find a and b such that the critical point is at (−3, 8). a = b = (c) For the values of a and b in part (b), is (−3, 8) a local maximum, local minimum, or a saddle point?
a.
[tex]f(x,y)=e^{-(x-a)^2-(y-b)^2}\implies\begin{cases}f_x=-2(x-a)e^{-(x-a)^2-(y-b)^2}\\f_y=-2(y-b)e^{-(x-a)^2-(y-b)^2}\end{cases}[/tex]
Critical points occur where [tex]f_x=f_y=0[/tex]. The exponential factor is always positive, so we have
[tex]\begin{cases}-2(x-a)=0\\-2(y-b)=0\end{cases}\implies(x,y)=\boxed{(a,b)}[/tex]
b. As the previous answer established, the critical point occurs at (-3, 8) if [tex]\boxed{a=-3}[/tex] and [tex]\boxed{b=8}[/tex].
c. Check the determinant of the Hessian matrix of [tex]f(x,y)[/tex]:
[tex]\mathbf H(x,y)=\begin{bmatrix}f_{xx}&f_{xy}\\f_{yx}&f_{yy}\end{bmatrix}[/tex]
The second-order derivatives are
[tex]f_{xx}=(-2+4(x-a)^2)e^{-(x-a)^2-(y-b)^2}[/tex]
[tex]f_{xy}=4(x-a)(y-b)e^{-(x-a)^2-(y-b)^2}[/tex]
[tex]f_{yx}=4(x-a)(y-b)e^{-(x-a)^2-(y-b)^2}[/tex]
[tex]f_{yy}=(-2+4(y-b)^2)e^{-(x-a)^2-(y-b)^2}[/tex]
so that the determinant of the Hessian is
[tex]\det\mathbf H(x,y)=f_{xx}f_{yy}-{f_{xy}}^2=\left((4(x-a)^2-2)(4(y-b)^2-2)-16(x-a)^2(y-b)^2\right)e^{-2(x-a)^2-2(y-b)^2}[/tex]
[tex]\det\mathbf H(x,y)=(16(x-a)^2(y-b)^2-8(x-a)^2-8(y-b)^2)+4)e^{-2(x-a)^2-2(y-b)^2}[/tex]
The sign of the determinant is unchanged by the exponential term so we can ignore it. For [tex]a=x=-3[/tex] and [tex]b=y=8[/tex], the remaining factor in the determinant has a value of 4, which is positive. At this point we also have
[tex]f_{xx}(-3,8;a=-3,b=8)=-2[/tex]
which is negative, and this indicates that (-3, 8) is a local maximum.
The critical point of the given function is (a, b). For it to be at (-3, 8), a and b should be -3 and 8 respectively. This point (-3, 8) is a local maximum for the function.
Explanation:The critical point of the given equation f(x, y) = e−(x − a)² − (y − b)² are the coordinates (a, b).
To find the values of a and b such that the critical point is at (−3, 8), we simply set (a, b) = (−3, 8). This implies that a = -3 and b = 8.In the context of the function f(x, y), the given point (-3,8) represents a local maximum. This is because the function f(x, y) achieves its maximum value when its exponent is at a minimum, which occurs at x=a and y=b. Hence we can say that (-3, 8) is a local maximum for the function f(x, y) with a = -3 and b = 8.Learn more about Critical points here:https://brainly.com/question/32077588
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Evaluate the surface integral ∫∫s F · dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F(x, y, z) = x i + y j + 10 k. S is the boundary of the region enclosed by the cylinder x2 + z2 = 1 and the planes y = 0 and x + y = 8.
Use the divergence theorem.
[tex]\vec F(x,y,z)=x\,\vec\i\math+y\,\vec\jmath+10\,\vec k\implies\nabla\cdot\vec F(x,y,z)=\dfrac{\partial(x)}{\partial x}+\dfrac{\partial(y)}{\partial y}+\dfrac{\partial(10)}{\partial z}=2[/tex]
The div theorem says the integral of [tex]\vec F[/tex] across [tex]S[/tex] is equal to the integral of [tex]\nabla\cdot\vec F[/tex] over the region with boundary [tex]S[/tex] (call it [tex]R[/tex]):
[tex]\displaystyle\iint_S\vec F(x,y,z)\cdot\mathrm d\vec S=\iiint_R(\nabla\cdot\vec F(x,y,z))\,\mathrm dx\,\mathrm dy\,\mathrm dz[/tex]
[tex]=\displaystyle2\iiint_R\mathrm dx\,\mathrm dy\,\mathrm dz[/tex]
Convert to cylindrical coordinates:
[tex]\begin{cases}x=r\cos\theta\\y=y\\z=r\sin\theta\end{cases}\implies\mathrm dx\,\mathrm dy\,\mathrm dz=r\,\mathrm dr\,\mathrm d\theta\,\mathrm dy[/tex]
Then the integral is
[tex]=\displaystyle2\int_0^{2\pi}\int_0^1\int_0^{8-r\cos\theta}r\,\mathrm dy\,\mathrm dr\,\mathrm d\theta=\boxed{16\pi}[/tex]
5x + 2 = 12
Solve the following equation. Then place the correct number in the box provided.
Answer:
x = 2
Step-by-step explanation:
5x + 2 = 12
Subtract 2 from both sides
5x + 2 - 2 = 12 - 2
Simplifying
5x = 10
Divide both sides by 5
5x / 5 = 10/5
Simplifying
x = 2
Answer:
x = 2
Step-by-step explanation:
You are solving for the variable, x. Note the equal sign, what you do to one side, you do to the other. Do the opposite of PEMDAS, and isolate the variable.
First, subtract 2 from both sides.
5x + 2 = 12
5x + 2 (-2) = 12 (-2)
5x = 12 - 2
5x = 10
Isolate the variable, x. Divide 5 from both sides.
(5x)/5 = (10)/5
x = 10/5
x = 2
x = 2 is your answer.
~
p 3 • p 2 • p
Multiply or divide as indicated.
heeelllpppp
For this case we have the following expression:
[tex]p ^ 3 * p ^ 2 * p =[/tex]
By definition of multiplication of powers of the same base, we have to put the same base and add the exponents, that is:
[tex]a ^ n * a ^ m = a ^ {n + m}[/tex]
So:
[tex]p ^ 3 * p ^ 2 * p = p^{ 3 + 2 + 1} = p^6[/tex]
Answer:
[tex]p^6[/tex]
2x • 3x 2
Multiply or divide as indicated.
For this case we have the following expression:
[tex]2x * 3x ^ 2 =[/tex]
By definition of multiplication of powers of the same base, we have to put the same base and add the exponents, that is:
[tex]a ^ n * a ^ m = a ^ {n + m}[/tex]
So:
[tex]2x * 3x ^ 2 = (2 * 3) x^{1 + 2} = 6x ^ 3[/tex]
Answer:
[tex]6x ^ 3[/tex]
Answer: [tex]6x^3[/tex]
Step-by-step explanation:
You need to remember a property called "Product of powers property". This property states the following:
[tex](a^m)(a^n)=a^{(m+n)}[/tex]
In this case you can observe that the bases of the expression [tex]2x*3x^2[/tex] are equal, this means that you can apply the property mentioned before.
Knowing this you can make the multiplication.
Then, the product you get is:
[tex]2x*3x^2=6x^{(1+2)}=6x^3[/tex]
(Please Hurry!) Which expression is equivalent to the following complex fraction?
Answer:
The expression equivalent to the given complex fraction is
[tex]\frac{-2x+5y}{3x-2y}[/tex]
Step-by-step explanation:
An easy way to solve the complex fraction is to solve the numerator and denominator separately.
Numerator:
[tex]\frac{-2}{x} + \frac{5}{y}\\ = \frac{-2y + 5x}{xy}[/tex]
Denominator:
[tex]\frac{3}{y} + \frac{-2}{x}\\ = \frac{3x - 2y}{xy}[/tex]
Solving the complex fraction:
[tex][\frac{-2}{x} + \frac{5}{y}] / [\frac{3}{y} + \frac{-2}{x}]\\= [\frac{-2y + 5x}{xy}] / [\frac{3x - 2y}{xy}][/tex]
[tex]=\frac{-2y + 5x}{xy} * \frac{xy}{3x - 2y}[/tex]
Common terms in the numerator and denominator cancels each other(Cross multiplication) :
[tex]= \frac{-2y + 5x}{3x - 2y}[/tex]
Solve the following system of equations
-4x - 9y =24
7x + 3y =9
Answer:
The solution is:
[tex](3, -4)[/tex]
Step-by-step explanation:
We have the following equations
[tex]-4x - 9y =24[/tex]
[tex]7x + 3y =9[/tex]
To solve the system multiply by 3 the second equation and add it to the first equation
[tex]3*7x + 3*3y =3*9[/tex]
[tex]21x + 9y =27[/tex]
[tex]-4x - 9y =24[/tex]
---------------------------------------
[tex]17x=51[/tex]
[tex]x=\frac{51}{17}[/tex]
[tex]x=3[/tex]
Now substitute the value of x in any of the two equations and solve for y
[tex]7(3) + 3y =9[/tex]
[tex]21 + 3y =9[/tex]
[tex]3y =9-21[/tex]
[tex]3y =-12[/tex]
[tex]y =-\frac{12}{3}[/tex]
[tex]y =-4[/tex]
The solution is:
[tex](3, -4)[/tex]
Answer:
x = 3 and y = -4
Step-by-step explanation:
It is given that,
-4x - 9y = 24 -----(1)
7x + 3y = 9 ---(2)
To find the solution of given equations
eq(2) * 3 ⇒
21x + 9y = 27 -----(3)
eq(1) + eq(3) ⇒
-4x - 9y = 24 -----(1)
21x + 9y = 27 -----(3)
17x = 51
x = 51/17 = 3
Substitute the value of x in eq(1)
-4x - 9y = 24 -----(1)
-4*3 - 9y = 24
-9y = 24 + 12
-9y = 36
y = 36/(-9) = -4
Therefore x = 3 and y = -4
A scientist has two solutions, which she has labeled Solution A and Solution B. Each contains salt. She knows that Solution A is 70% salt and Solution B is 95% salt. She wants to obtain 150 ounces of a mixture that is 75% salt. How many ounces of each solution should she use?
Answer:
Solution A: 120 ounces
Solution B: 30 ounces
Step-by-step explanation:
Let's call A the amount of Solution A. Solution A is 70% salt
Let's call B the amount of Solution B. Solution A is 95% salt
The resulting mixture should have 75% salt and 150 ounces .
Then we know that the total amount of mixture will be:
[tex]A + B = 150[/tex]
Then the total amount of salt in the mixture will be:
[tex]0.7A + 0.95B = 0.75 * 150[/tex]
[tex]0.7A + 0.95B = 112.5[/tex]
Then we have two equations and two unknowns so we solve the system of equations. Multiply the first equation by -0.95 and add it to the second equation:
[tex]-0.95A -0.95B = 150 * (-0.95)[/tex]
[tex]-0.95A -0.95B =-142.5[/tex]
[tex]-0.95A -0.95B =-142.5[/tex]
+
[tex]0.7A + 0.95B = 112.5[/tex]
--------------------------------------
[tex]-0.25A = -30[/tex]
[tex]A = \frac{-30}{-0.25}[/tex]
[tex]A = 120\ ounces[/tex]
We substitute the value of A into one of the two equations and solve for B.
[tex]120 + B = 150[/tex]
[tex]B = 30\ ounces[/tex]
A mass weighing 1 slug is attached to the end of a spring with spring constant 26 lb/ft. Damping force is numerically equal to 2 times the velocity. The mass is pulled down 6 ft and then released from rest. Write down and solve the initial value problem for the position x(t)
Answer:
x = e^(-t) [1.2 sin(5t) + 6 cos(5t)]
Step-by-step explanation:
A damped oscillator has the equation of motion:
m d²x/dt² + β dx/dt + k x = 0
This is an example of a second order linear ordinary differential equation with constant coefficients.
d²x/dt² + b dx/dt + c x = 0
Notice the leading coefficient is 1.
If b²−4c > 0, then the solution is:x = C₁ e^(-½ t (b + √(b²−4c) )) + C₂ e^(-½ t (b − √(b²−4c) ))
If b²−4c = 0, then the solution is:x = (C₁ t + C₂) e^(-bt/2)
If b²−4c < 0, then the solution is:x = e^(-bt/2) [C₁ sin(½ t √(4c−b²)) + C₂ cos(½ t √(4c−b²))]
Given that m = 1, β = 2, and k = 26:
d²x/dt² + 2 dx/dt + 26 x = 0
Here, b = 2 and c = 26, so:
b²−4c = (2)²−4(26) = -100 < 0
The general solution is:
x = e^(-2t/2) [C₁ sin(½ t √100) + C₂ cos(½ t √100)]
x = e^(-t) [C₁ sin(5t) + C₂ cos(5t)]
To find the values of C₁ and C₂, first find dx/dt, then plug in the initial conditions.
dx/dt = e^(-t) [5C₁ cos(5t) − 5C₂ sin(5t)] − e^(-t) [C₁ sin(5t) + C₂ cos(5t)]
dx/dt = e^(-t) [5C₁ cos(5t) − 5C₂ sin(5t) − C₁ sin(5t) − C₂ cos(5t)]
dx/dt = e^(-t) [(5C₁ − C₂) cos(5t) − (5C₂ + C₁) sin(5t)]
Given x(0) = 6:
6 = e^(0) [C₁ sin(0) + C₂ cos(0)]
6 = C₂
Given x'(0) = 0:
0 = e^(0) [(5C₁ − C₂) cos(0) − (5C₂ + C₁) sin(0)]
0 = 5C₁ − C₂
0 = 5C₁ − 6
C₁ = 1.2
So the solution is:
x = e^(-t) [1.2 sin(5t) + 6 cos(5t)]
Here's the graph:
desmos.com/calculator/bavfsoju5c
11. A sample of n = 25 scores has a mean of M = 68. Find the z-score for this sample: a. If it was obtained from a population with μ = 60 and σ = 10. b. If it was obtained from a population with μ = 60 and σ = 20. c. If it was obtained from a population with μ = 60 and σ = 40. 12. A population forms Gravetter, Frederick J. Statistics for The Behavioral Sciences (p. 221). Cengage Learning. Kindle Edition.
Answer:
a) z-score = 4
b) z-score = 2
c) z-score = 1
Step-by-step explanation:
* Lets revise some definition to solve the problem
- The mean of the distribution of sample means is called M
- The standard deviation of the distribution of sample means is
called σM (standard error)
- σM = σ/√n , where σ is the standard deviation and n is the sample size
- z-score = (M - μ)/σM, where μ is the mean of the population
* Lets solve the problem
∵ The sample size n = 25
∵ The sample mean M = 68
a)
∵ The mean of population μ = 60
∵ The standard deviation σ = 10
- Lets find σM to find z-score
∵ σM = σ/√n
∴ σM = 10/√25 = 10/5 = 2
- Lets find z-score
∵ z-score = (M - μ)/σM
∴ z-score = (68 - 60)/2 = 8/2 = 4
* z-score = 4
b)
∵ The mean of population μ = 60
∵ The standard deviation σ = 20
- Lets find σM to find z-score
∵ σM = σ/√n
∴ σM = 20/√25 = 20/5 = 4
- Lets find z-score
∵ z-score = (M - μ)/σM
∴ z-score = (68 - 60)/4 = 8/4 = 2
* z-score = 2
c)
∵ The mean of population μ = 60
∵ The standard deviation σ = 40
- Lets find σM to find z-score
∵ σM = σ/√n
∴ σM = 40/√25 = 40/5 = 8
- Lets find z-score
∵ z-score = (M - μ)/σM
∴ z-score = (68 - 60)/8 = 8/8 = 1
* z-score = 1
The z-score for a sample is calculated using the formula z = (x - μ) / σ, where x is the sample mean, μ is the population mean, and σ is the population standard deviation. For the given scenarios, the z-scores are 0.8, 0.4, and 0.2, respectively.
Explanation:To find the z-score for a sample, we use the formula:
z = (x - μ) / σ
where x is the sample mean, μ is the population mean, and σ is the population standard deviation.
a. If it was obtained from a population with μ = 60 and σ = 10:
z = (68 - 60) / 10 = 0.8
b. If it was obtained from a population with μ = 60 and σ = 20:
z = (68 - 60) / 20 = 0.4
c. If it was obtained from a population with μ = 60 and σ = 40:
z = (68 - 60) / 40 = 0.2
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What is -8/25(x-25)^2 + 32 in the standard form?
Answer:
(-8 (x + -35) (x - 15))/25
Step-by-step explanation:
Simplify the following:
32 - (8 (x - 25)^2)/25
Hint: | Put the fractions in 32 - (8 (x - 25)^2)/25 over a common denominator.
Put each term in 32 - (8 (x - 25)^2)/25 over the common denominator 25: 32 - (8 (x - 25)^2)/25 = 800/25 - (8 (x - 25)^2)/25:
800/25 - (8 (x - 25)^2)/25
Hint: | Combine 800/25 - (8 (x - 25)^2)/25 into a single fraction.
800/25 - (8 (x - 25)^2)/25 = (800 - 8 (x - 25)^2)/25:
(800 - 8 (x - 25)^2)/25
Hint: | Factor out the greatest common divisor of the coefficients of 800 - 8 (x - 25)^2.
Factor -8 out of 800 - 8 (x - 25)^2:
(-8 ((x - 25)^2 - 100))/25
Hint: | Write 100 as a square in order to express (x - 25)^2 - 100 as a difference of squares.
(x - 25)^2 - 100 = (x - 25)^2 - 10^2:
(-8 ((x - 25)^2 - 10^2))/25
Hint: | Factor the difference of two squares.
Factor the difference of two squares. (x - 25)^2 - 10^2 = ((x - 25) - 10) ((x - 25) + 10):
(-8(x - 25 - 10) (x - 25 + 10))/25
Hint: | Group like terms in x - 25 + 10.
Grouping like terms, x - 25 + 10 = x + (10 - 25):
(-8 x + (10 - 25) (x - 25 - 10))/25
Hint: | Evaluate 10 - 25.
10 - 25 = -15:
(-8 (x - 25 - 10) (x + -15))/25
Hint: | Group like terms in x - 25 - 10.
Grouping like terms, x - 25 - 10 = x + (-25 - 10):
(-8 x + (-25 - 10) (x - 15))/25
Hint: | Evaluate -25 - 10.
-25 - 10 = -35:
Answer: (-8 (x + -35) (x - 15))/25