Answer:
The new mean = 3 × (the old mean) = 150
The new standard deviation is also = 3 × (The old standard deviation) = 15
Step-by-step explanation:
µ = 50 and σ = 5
The mean is the sum of variables divided by the number of variables.
Mean = (Σx)/N = µ = 50
x = each variable
N = number of variables
If each variable changed to 3x
Mean becomes
Mean = (Σ(3x))/N = 3 (Σx)/N = 3 × µ = 3 × 50 = 150.
The standard deviation is the square root of variance. And variance is an average of the squared deviations from the mean.
The standard deviation measures the rate of spread of the data set around the mean.
Standard deviation = σ = √[Σ(x - µ)²/N]
x = each variable
µ = mean
N = number of variables
If each variable changed to 3x
Recall µ changed to 3µ
Standard deviation = σ = √[Σ(3x - 3µ)²/N]
σ = √[Σ 3² (x - µ)²/N] = √[(3²)Σ(x - µ)²/N] = 3×√[Σ(x - µ)²/N] = 3 × σ = 3 × 5 = 15
If every score is multiplied by 3, it is logical to reason that the average of the new set of numbers also is 3× the old average.
And the new set of numbers spread out similarly around this new mean, only that the new space of spread is now 3× the old one.
When every score in a population is multiplied by a constant, both the mean and the standard deviation are multiplied by that constant. So, in the given example, the new mean would be 150 and the new standard deviation would be 15.
Explanation:In mathematics, when every score in a population is multiplied by a constant, both the mean (µ) and the standard deviation (σ) are also multiplied by that constant.
So, if every score in a population with µ = 50 and σ = 5 is multiplied by 3, the new mean (µ_new) would be 50 * 3 = 150, and the new standard deviation (σ_new) would be 5 * 3 = 15.
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Please help meeeeeeeeeeeee
Answer: the solutions of the equation are
x = - 1
x = 8
x = 9
Step-by-step explanation:
The given cubic equation is expressed as
x³ - 16x² + 55x + 72 = 0
The first step is to test for any value of x that satisfies the equation when
x³ - 16x² + 55x + 72 = 0
Assuming x = - 1, then
- 1³ - 16(-1)³ + 55(-1) + 72 = 0
- 1 - 16 - 55 + 72 = 0
0 = 0
It means that x + 1 is a factor.
To determine the other factors, we would apply the long division method. The steps are shown in the attached photo. Looking at the photo, we would factorize the quadratic equation which is expressed as
x² - 17x + 72 = 0
x² - 9x - 8x + 72 = 0
x(x - 9) - 8(x - 9) = 0
(x - 9)(x - 8) = 0
I WILL GIVE A CROWN JUST NEED HELP ASAP
Answer:
C option is correct 5/8.
Step-by-step explanation:
Ela ate chocolate on Tuesday = 1/8
Remaining chocolate = 1 - 1/8
= 7/8
Ela ate chocolate on Wednesday = 7/8
Chocolate left = 7/8 - 2/8
= 5/8
Find an equation of the line. Write the equation in standard form.
Horizontal; through (3,-9)
Answer:
y = -9
Step-by-step explanation:
Standard form of the equation of a line is ...
ax + by = c
When that line is a horizontal line, this can be reduced to ...
y = c
The value of c must be the same as the y-coordinate of the point you want this line to go through.
y = -9 . . . . . . horizontal line through (3, -9)
solve for x.
A. 13
B. 14
C. 12
D. 11
Answer:
the answer is a
Step-by-step explanation:
Answer:
x = 13
Step-by-step explanation:
One of the THEOREM for triangles states that, A LINE DRAWN PARALLEL TO ONE SIDE OF THE TRIANGLE DIVIDES THE OTHER TWO SIDES IN THESAME RATIO.
From the figure above:
VW is parallel to SU.
VW divides ST and TU in thesame ration.
Hence:
TV / SV = TW / UW
14 / 6 = 21 / (x - 4)
Cross Multiplying gives:
14( x - 4) = 6 * 21
14x - 56 = 126
14x = 126 + 56
14x = 182
Divide through by 14.
x = 182/14
x = 13
If Henry were to add 5 gallons of water to a tank that is already 3434 full of water, the tank would be 7878 full. How many gallons of water would the tank hold if it were full
Answer:
The tank would hold 40 gallons of water when full.
Step-by-step explanation:
The tank is already 3/4 full of water
If Henry added 5 gallons, it will be 7/8 full.
Therefore the fraction added by Henry =7/8-3/4=1/8
It means that 1/8 of the total=5 gallons of water
Let the total capacity of the tank=x
[tex]\frac{1}{8}Xx=5 gallons\\ x=8X5=40 gallons[/tex]
The tank would hold 40 gallons of water when full.
A map is shown with a scale drawing of 1 inch = 15 miles,nicloe measured the distance to the next town as 3 inches. How many miles does she have to travel to get to the next town
Answer: There are 45 miles that she have to travel to get to the next down.
Step-by-step explanation:
Since we have given that
1 inch = 15 miles
If there are given 3 inches.
We need to find the number of miles she have to travel to get to travel to get to the next town.
So, it becomes,
[tex]\dfrac{1}{15}=\dfrac{3}{x}\\\\x=15\times 3\\\\x=45\ miles[/tex]
Hence, there are 45 miles that she have to travel to get to the next down.
Which BEST describes the system of equations?
x = y -3 - 4x =12 - 4y
Consistent
Inconsistent
Consistent and Dependent
D)
Consistent and Independent
Answer:
Hi, the correct answer to this would be B, INCONSISTENT.
I researched this and my cite is www.quizlet.com. It is this answer because it is not consistent Nor C,D.
Step-by-step explanation:
I answered this before but someone deleted every single question I ever answered. Hahaha
Hope this helps :)
Answer:
The answer is CONSISTENT AND DEPENDENT
Step-by-step explanation:
USA Test Prep
Points N and R both lie on circle O. Line segment RQ is tangent to the circle at point R.
What is the perimeter of triangle RON?
Answer:
Perimeter ΔRON = 18.66 units
Step-by-step explanation:
Complete Question has these information given as well:
Circle with center ORN = QNRQ = [tex]5\sqrt{3}[/tex]ON = 5 unitsRQ is tangentNow, we draw an image according to the information given. THe image drawn is attached.
Looking at the figure, we can say the perimeter of triangle RON would be:
RON = OR + ON + RN
We know according to tangent theorem, QN = QR
RN is equal to that as well and RN = [tex]5\sqrt{3}[/tex]
Now,
OR and ON is the radius, which is "5"
Perimeter of RON = 5 + 5 + [tex]5\sqrt{3}[/tex] = 10 + [tex]5\sqrt{3}[/tex] = 18.66 units
Perimeter ΔRON = 18.66 units
Answer:
18.7 units
Step-by-step explanation:
Which statement describes the system of equations?
It has infinitely many solutions.
It has no solution.
It has one solution .
It has one solution (8, 2).
Answer:
Step-by-step explanation:
it has no solution (8,2)
just took the test
what is the expansion of (3+x)^4
Answer:
[tex]\left(3+x\right)^4:\quad x^4+12x^3+54x^2+108x+81[/tex]
Step-by-step explanation:
Considering the expression
[tex]\left(3+x\right)^4[/tex]
Lets determine the expansion of the expression
[tex]\left(3+x\right)^4[/tex]
[tex]\mathrm{Apply\:binomial\:theorem}:\quad \left(a+b\right)^n=\sum _{i=0}^n\binom{n}{i}a^{\left(n-i\right)}b^i[/tex]
[tex]a=3,\:\:b=x[/tex]
[tex]=\sum _{i=0}^4\binom{4}{i}\cdot \:3^{\left(4-i\right)}x^i[/tex]
Expanding summation
[tex]\binom{n}{i}=\frac{n!}{i!\left(n-i\right)!}[/tex]
[tex]i=0\quad :\quad \frac{4!}{0!\left(4-0\right)!}3^4x^0[/tex]
[tex]i=1\quad :\quad \frac{4!}{1!\left(4-1\right)!}3^3x^1[/tex]
[tex]i=2\quad :\quad \frac{4!}{2!\left(4-2\right)!}3^2x^2[/tex]
[tex]i=3\quad :\quad \frac{4!}{3!\left(4-3\right)!}3^1x^3[/tex]
[tex]i=4\quad :\quad \frac{4!}{4!\left(4-4\right)!}3^0x^4[/tex]
[tex]=\frac{4!}{0!\left(4-0\right)!}\cdot \:3^4x^0+\frac{4!}{1!\left(4-1\right)!}\cdot \:3^3x^1+\frac{4!}{2!\left(4-2\right)!}\cdot \:3^2x^2+\frac{4!}{3!\left(4-3\right)!}\cdot \:3^1x^3+\frac{4!}{4!\left(4-4\right)!}\cdot \:3^0x^4[/tex]
[tex]=\frac{4!}{0!\left(4-0\right)!}\cdot \:3^4x^0+\frac{4!}{1!\left(4-1\right)!}\cdot \:3^3x^1+\frac{4!}{2!\left(4-2\right)!}\cdot \:3^2x^2+\frac{4!}{3!\left(4-3\right)!}\cdot \:3^1x^3+\frac{4!}{4!\left(4-4\right)!}\cdot \:3^0x^4[/tex]
as
[tex]\frac{4!}{0!\left(4-0\right)!}\cdot \:\:3^4x^0:\:\:\:\:\:\:81[/tex]
[tex]\frac{4!}{1!\left(4-1\right)!}\cdot \:3^3x^1:\quad 108x[/tex]
[tex]\frac{4!}{2!\left(4-2\right)!}\cdot \:3^2x^2:\quad 54x^2[/tex]
[tex]\frac{4!}{3!\left(4-3\right)!}\cdot \:3^1x^3:\quad 12x^3[/tex]
[tex]\frac{4!}{4!\left(4-4\right)!}\cdot \:3^0x^4:\quad x^4[/tex]
so equation becomes
[tex]=81+108x+54x^2+12x^3+x^4[/tex]
[tex]=x^4+12x^3+54x^2+108x+81[/tex]
Therefore,
[tex]\left(3+x\right)^4:\quad x^4+12x^3+54x^2+108x+81[/tex]Suppose that a brand of lightbulb lasts on average 1730 hours with a standard deviation of 257 hours. Assume the life of the lightbulb is normally distributed. Calculate the probability that a particular bulb will last from 1689 to 2267 hours?
Answer:
P [ 1689 ≤ X ≤ 2267 ] = 54,88 %
Step-by-step explanation:
Normal Distribution
Mean μ₀ = 1730
Standard Deviation σ = 257
We need to calculate z scores for the values 1689 and 2267
We apply formula for z scores
z = ( X - μ₀ ) /σ
X = 1689 then
z = (1689 - 1730)/ 257 ⇒ z = - 41 / 257
z = - 0.1595
And from z table we get for z = - 0,1595
We have to interpolate
- 0,15 0,4364
- 0,16 0,4325
Δ = 0.01 0.0039
0,1595 - 0,15 = 0.0095
By rule of three
0,01 0,0039
0,0095 x ?? x = 0.0037
And 0,4364 - 0.0037 = 0,4327
Then P [ X ≤ 1689 ] = 0.4327 or P [ X ≤ 1689 ] = 43,27 %
And for the upper limit 2267 z score will be
z = ( X - 1730 ) / 257 ⇒ z = 537 / 257
z = 2.0894
Now from z table we find for score 2.0894
We interpolate and assume 0.9815
P [ X ≤ 2267 ] = 0,9815
Ths vale already contains th value of P [ X ≤ 1689 ] = 0.4327
Then we subtract to get 0,9815 - 0,4327 = 0,5488
Finally
P [ 1689 ≤ X ≤ 2267 ] = 0,5488 or P [ 1689 ≤ X ≤ 2267 ] = 54,88 %
A group of equations that have a common intersection point is called
Answer:
system of equations i think
Step-by-step explanation:
A group of equations that have a common intersection point is called: system of equations.
A system of equations can be defined as an algebraic equation of the first order with two (2) variables and each of its term having an exponent of one (1).
Generally, a system of equations in two (2) variables must have at least two (2) solution.
This ultimately implies that, a system of equations must have a common intersection point.
In Mathematics, an example of a system of equations include the following:
[tex]2x + 4y = 8[/tex] ....equation 1.[tex]2x - 8y = 20[/tex] ....equation 2.Additionally, the above system of equations can easily be solved by using an elimination method.
In conclusion, a solution of the group of equations is an ordered pair that satisfies all the equations in a system of equations.
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Invest $6,300 in two different accounts the first account paid 11% the second account paid 6% in interest at the end of the year he had earned $643 in interest how much was in each account
Answer: he invested $5300 at 11% and $1000 at 6%
Step-by-step explanation:
Let x represent the amount which he invested in the first account paying 11% interest.
Let y represent the amount which he invested in the second account paying 6% interest.
He Invest $6,300 in two different accounts the first account paid 11% the second account paid 6% in interest. This means that
x + y = 6300
The formula for determining simple interest is expressed as
I = PRT/100
Considering the first account paying 11% interest,
P = $x
T = 1 year
R = 11℅
I = (x × 11 × 1)/100 = 0.11x
Considering the second account paying 6% interest,
P = $y
T = 1 year
R = 6℅
I = (y × 6 × 1)/100 = 0.06y
At the end of the year, he had earned $643 in interest , it means that
0.11x + 0.06y = 643 - - - - - - - - - -1
Substituting x = 6300 - y into equation 1, it becomes
0.11(6300 - y) + 0.06y = 643
693 - 0.11y + 0.06y = 643
- 0.11y + 0.06y = 643 - 693
- 0.05y = - 50
y = - 50/ - 0.05
y = 1000
x = 6300 - y = 6300 - 1000
x = 5300
Complete the equivalent equation for –7x – 60 = x2 + 10x.
(x + )(x + ) = 0
What are the solutions of –7x – 60 = x2 + 10x?
x =
The equivalent equation is [tex](x+5)(x+12)=0[/tex]
The solution are [tex]x=-5, x=-12[/tex]
Explanation:
Given that the equation is [tex]-7 x-60=x^2+10 x[/tex]
Simplifying the equation, we get,
[tex]0=x^2+10 x+7x+60[/tex]
Switch sides, we have,
[tex]x^2+17 x+60=0[/tex]
Equivalent equation:
Let us factor the quadratic equation.
Thus, we have,
[tex]x^{2} +5x+12x+60=0[/tex]
Grouping the terms, we get,
[tex]x(x+5)+12(x+5)=0[/tex]
Factoring out (x+5), we get,
[tex](x+5)(x+12)=0[/tex]
Thus, the equivalent equation is [tex](x+5)(x+12)=0[/tex]
Solution:
Solving the equation [tex](x+5)(x+12)=0[/tex], we get,
[tex]x+5=0[/tex] and [tex]x+12=0[/tex]
[tex]x=-5[/tex] and [tex]x=-12[/tex]
Thus, the solutions are [tex]x=-5[/tex] and [tex]x=-12[/tex]
Answer:
A. 5
B. 12
C. -12 or -5
Step-by-step explanation:
(x + 5)(x + 12) = 0
What are the solutions of –7x – 60 = x2 + 10x?
x = -12 or -5
50 POINTS AND BRAINLIEST!!
Drag the expressions into the boxes to correctly complete the table.
Answer:
View Image
Step-by-step explanation:
To identify if it's a polynomial, look at the x and its exponent.
x CANNOT be:
1. In the denominator: [tex]\frac{1}{x}[/tex] NOT polynomial
2. In the exponent: [tex]2^x[/tex] NOT polynomial
3. In a root: [tex]\sqrt{x}[/tex] NOT polynomial
The exponent on the x must be a positive integer, therefore,
exponent:
1.) Cannot be a fraction: [tex]x^{1/2}[/tex] NOT polynomial
2.) Cannot be negative: [tex]x^{-2}[/tex] NOT polynomial
Find the missing height of the parallelogram when the area is 340 cm².
Answer:
17 cm
Step-by-step explanation:
The formula for the area of a parallelogram is ...
A = bh
The base (b) is given as (5 cm +15 cm) = 20 cm. The area is given as 340 cm^2. Filling in the give numbers, we have ...
340 cm^2 = (20 cm)h
Dividing by the coefficient of h gives ...
(340 cm^2)/(20 cm) = h = 17 cm
The height of the parallelogram is 17 cm.
_____
Comment on the geometry
The triangles at either end of the figure will be 5-12-13 right triangles, meaning the height of the figure is 12 cm. Using that height, we find the area to be (12 cm)(20 cm) = 240 cm^2 (not 340 cm^2). This leads us to believe there is a typo in the problem statement.
As the problem is given, the geometry is impossible. The height of the parallelogram cannot be greater than the length of the slanted side.
As the problem is written, it is a "one-step" problem. (Divide the area by the base length.) If the area were not given, then the Pythagorean theorem would be required to find the height. That is a 2-step problem:
13^2 = 5^2 + h^2
h^2 = 169 -25 = 144 . . . . solve for h²
h = √144 = 12 . . . . . . . . . take the square root
It is not at all clear what 3 steps you're supposed to show in your work.
How do you writer this function in standard form??
Question 1) Function defining the table:
From the table the x-intercepts are -2 and 1. This means the factors are:
(x+2) and (x-1)
Let
[tex]h(x) = a(x + 2)(x - 1)[/tex]
The point (-1,-1) satisfy this function since it is from the same table.
[tex] - 1 = a( - 1 + 2)( - 1 - 1) \\ - 1 = - 2a \\ a = \frac{1}{2} [/tex]
Therefore the function is
[tex]h(x) = \frac{1}{2} (x + 2)(x - 1)[/tex]
We expand to get:
[tex]h(x) = \frac{1}{2} ( {x}^{2} + x - 2)[/tex]
The standard form is:
[tex]h(x) = \frac{ {x}^{2} }{2} + \frac{x}{2} - 1[/tex]
Question 3) Parabola opening up
The x-intercepts are x=3 and x=7
The factors are (x-3), (x-7)
The factored from is
[tex]y = a(x - 3)(x - 7)[/tex]
The curve passes through (5,-4)
[tex] - 4= a( 5- 3)( 5 - 7) \\ - 4= - 4a \\ a = 1[/tex]
The equation is:
[tex]y = (x - 3)(x + 7)[/tex]
Expand:
[tex]y = {x}^{2} + 7x - 3x - 21[/tex]
[tex]y = {x}^{2} + 4x - 21[/tex]
This is the standard form:
Question 3) Parabola opening down:
The x-intercepts are x=-5 and x=1
The factors are (x+5), (x-1)
The factored form is
[tex]y = - (x + 5)(x - 1)[/tex]
We expand to get:
[tex]y = - ( {x}^{2} - x + 5x - 5)[/tex]
[tex]y = - {x}^{2} - 4x + 5[/tex]
This is the standard form.
The average wall thickness of 25 panes of glass is 4.05 mm. The standard deviation of the thickness of the 25 panes is measured to be 0.08 mm. What is the 90% confidence interval of the mean of wall thickness
Answer:
u => 4,028
Step-by-step explanation:
To find the answer, we have the following formula:
u => m - t (alpha, n-1) * [sd / (n) ^ (1/2)]
where m is the mean.
where sd is the standard deviation.
where n is the sample size.
t is a parameter that depends on the confidence interval and the sample size.
alpha = 1 - ci
ci = 90% = 0.9
Therefore, alpha = 1 - 0.9 = 0.1.
n - 1 = 25 - 1 = 24
So it would come being t (0.1, 24), if we look in the table, which I will attach the value of t is equal to 1.318.
We know the rest of the values, m = 4.05; sd = 0.08; n = 25
u => 4.05 - 1,318 * [0.08 / (25) ^ (1/2)]
u => 4.028
Which means that the interval with a 90% confidence of the wall thickness measurement is:
u => 4.028
SOMEONE HELP ME PLEASE I NEED AN EXPLANATION A GOOD ONE!!!!
AND ANSWER!!!!
Find the length of MK if JK=36, JL=48, and JN=60.
The length of MK is 9
Explanation:
The length of the sides are [tex]J K=36[/tex] , [tex]\mathrm{JL}=48[/tex] , [tex]J N=60[/tex]
We need to determine the length of MK
From the figure, we can see that JMN is a triangle and KL is parallel to MN.
Then, by side - splitter theorem, we have,
[tex]\frac{JK}{KM} =\frac{JL}{LN}[/tex]
where [tex]J K=36[/tex] , [tex]\mathrm{JL}=48[/tex]
The length of LN can be determined by subtracting JN and JL.
Thus, we have,
[tex]LN=JN-JL[/tex]
[tex]LN=60-48=12[/tex]
The length of LN is [tex]LN=12[/tex]
Substituting the values [tex]J K=36[/tex] , [tex]\mathrm{JL}=48[/tex] and [tex]LN=12[/tex] in [tex]\frac{JK}{KM} =\frac{JL}{LN}[/tex], we have,
[tex]\frac{36}{KM} =\frac{48}{12}[/tex]
Multiplying both sides by 12, we have,
[tex]\frac{36\times 12}{KM} =48[/tex]
[tex]\frac{432}{KM} =48[/tex]
[tex]\frac{432}{48} =KM[/tex]
[tex]9=KM[/tex]
Thus, the length of MK is 9
Write simplified expressions for the area and perimeter of the rectangle. Area: 8 8 x+ x+ Perimeter: 2 2 x+ x+
The simplified expressions for the area and perimeter of the rectangle are 16x and 4x, respectively.
To simplify the expressions for the area and perimeter of the rectangle, we can extract common factors:
Area: [tex]\(8(x + x) = 8 \times 2x = 16x\)[/tex]
Perimeter: [tex]\(2(x + x) = 2 \times 2x = 4x\)[/tex]
Therefore, the simplified expressions are:
Area: 16x
Perimeter: 4x
We sample the performance of each step, and find that each of them operates correctly about 95% of the time. What is the probability that a customer who visits three times will find our service satisfactory on all 3 visits?
Answer:
85%
Step-by-step explanation:
The probability of finding our satisfactory service in the three visits in a row would be the multiplication of the probability of each event.
The event is always the same. 95% of the service will be satisfactory. That is a probability of 95/100
Then the final probability would be:
(95/100) * (95/100) * (95/100) = 0.85
In other words, the probability that a customer who visits three times will find our service satisfactory is 85%
You return a DVD movie that was 5 days overdue including a previous unpaid balance of $2.50, your new balance is $7.75. How much is the daily fine for an overdue DVD?
Answer:
$ 1.05
Step-by-step explanation:
To solve the problem it is necessary to pose some equations, which are the following:
Debt DVD = New Balance - Old Balance
With this we will calculate the net value that only has to do with the debt of the DVD:
Debt DVD: 7.75 - 2.50 = 5.25
Now, the other equation is the value per day that generates a delay.
Debt DVD / # days
Replacing
5.25 / 5 = 1.05
Then the daily fine for a overdueDVD is $ 1.05
Harry got a pair of rain boots and a pair of tennis shoes. Kate got a pair of tennis shoes and sandals what fraction of the set of new shoes is the rain boots
Answer:
1/4 or 25%
Step-by-step explanation:
The set of new shoes is composed by all pairs bought by either Harry or Kate.
Harry bought a pair of rain boots and a pair of tennis shoes while Kate got a pair of tennis shoes and sandals, totaling 4 pairs. Out of those 4 pairs, only 1 pair are rain boots, the fraction corresponding to rain boots is:
[tex]f = \frac{1}{4}=25\%[/tex]
Rain boots are 1/4 of the set of new shoes.
A graph of the function g(x) = x^4-8x³+x²+42x has zeros at -2, 0, 3 and 7. What are the signs of the values between 0 and 3? Show algebraically how you know.
Answer:
The answer to your question is Positive
Step-by-step explanation:
Function
g(x) = x⁴ - 8x³ + x² + 42x
To know if the function is positive or negative in the interval (0, 3), look for two numbers between this interval and evaluate the function.
The numbers I chose were 1 and 2
- g(1) = (1)⁴ - 8(1)³ + (1)² + 42(1)
= 1 - 8 - 1 + 42
= + 36 positive
- g(2) = (2)⁴ - 8(2)³ + (2)² + 42(2)
= 16 - 64 + 4 + 84
= + 40
Conclusion
The function is positive in the interval (0, 3)
#4 find the value of x. Round answer to nearest tenth.
Answer:
Step-by-step explanation:
From the given right angle triangle,
The unknown side represents the hypotenuse of the right angle triangle.
With m∠40 as the reference angle,
x represents the adjacent side of the right angle triangle.
4 represents the opposite side of the right angle triangle.
To determine x, we would apply
the Tangent trigonometric ratio.
Sin θ = opposite side/hypotenuse. Therefore,
Tan 40 = 4/x
x = 4/Tan 40 = 4/0.839
x = 4.8
The focal points of an ellipse are (12, 0) and (−12, 0), and the point (12, 7) is on the ellipse. Find the points where this curve intersects the coordinate axes.
Answer:
x - ( 16 , 0 ) , ( -16 , 0 )
y - ( 0 , 4√7 ) , ( 0 , -4√7)
Step-by-step explanation:
Solution:
- The sum of the distances from a point on the ellipse to its foci is constant. You have both foci and a point, so you can find the sum of the distances.
-Then you can find the vertices since they are points on the ellipse on the x-axis whose sum of distances to the foci are that value.
- The 7 in y coordinate of (12,7) is the length of semi-latus rectum. Also c is 12:
c^2 = a^2 + b^2
Where, a: x-intercept
b: y-intercept
- The length of semi-latus rectum is given by:
b^2 = 7*a
- Substitute latus rectum expression in the first one we get:
c^2 = a^2 + 7a
a^2 + 7a - 144 = 0
( a - 16 ) * ( a - 9 ) = 0
a = +/- ( 16 )
- The y-intercept we will use latus rectum expression again:
b = +/- √(7*16)
b = +/- 4√7
- The intercepts are:
x - ( 16 , 0 ) , ( -16 , 0 )
y - ( 0 , 4√7 ) , ( 0 , -4√7)
Final answer:
An ellipse is a closed curve where the sum of the distances from any point on the curve to the two foci is constant. To find where this particular ellipse intersects the coordinate axes, we can set y = 0 to find the points of intersection with the x-axis and set x = 0 to find the points of intersection with the y-axis.
Explanation:
An ellipse is a closed curve where the sum of the distances from any point on the curve to the two foci is constant. The focal points of this ellipse are (12, 0) and (-12, 0). We know that the point (12, 7) lies on the ellipse. To find where this curve intersects the coordinate axes, we need to find the points where the ellipse intersects the x-axis and y-axis.
Intersecting the x-axis:
To find the points where the ellipse intersects the x-axis, we set y = 0 and solve for x. In this case, the coordinates of the intersection points will be (x, 0).
Plugging in y = 0 into the equation of the ellipse:
(x - 12)^2 / a^2 + (0 - 0)^2 / b^2 = 1
Simplifying this equation:
(x - 12)^2 / a^2 = 1
Since the ellipse is symmetric about the y-axis, the x-coordinates of the intersection points will have the same absolute value but opposite signs. So, we can solve for a single value of x and then take its negative to find the other intersection point.
(x - 12)^2 / a^2 = 1
x - 12 = a
x = a + 12
The coordinates of the intersection points on the x-axis are (a + 12, 0) and (-a - 12, 0).
Intersecting the y-axis:
To find the points where the ellipse intersects the y-axis, we set x = 0 and solve for y. In this case, the coordinates of the intersection points will be (0, y).
Plugging in x = 0 into the equation of the ellipse:
(0 - 12)^2 / a^2 + (y - 0)^2 / b^2 = 1
Simplifying this equation:
144 / a^2 + y^2 / b^2 = 1
Solving for y:
y^2 / b^2 = 1 - 144 / a^2
y^2 / b^2 = (a^2 - 144) / a^2
y^2 = b^2 * (a^2 - 144) / a^2
y = ± sqrt(b^2 * (a^2 - 144) / a^2)
The coordinates of the intersection points on the y-axis are (0, ± sqrt(b^2 * (a^2 - 144) / a^2)).
What is the correct answer to number 9? Please explain step by step.
Answer:
The answer to your question is Momentum = [tex]\frac{(x + 2)^{2}}{x - 3}[/tex]
Step-by-step explanation:
Data
mass = [tex]\frac{x^{2}+ 4x + 4 }{x^{2}- 9}[/tex]
velocity = [tex]\frac{x^{2}+ 5x + 6}{x + 2}[/tex]
Formula
Momentum = mass x velocity
Substitution
Momentum = [tex]\frac{x^{2}+ 4x + 4}{x^{2}- 9} \frac{x^{2} + 5x + 6}{x + 2}[/tex]
Factor The first numerator is a perfect square trinomial and the second one is a trinomial of the form x² + bx + c.
Momentum = [tex]\frac{(x + 2)^{2}}{(x - 3)(x + 3)} \frac{(x + 2)(x + 3)}{x + 2}[/tex]
Simplify and result
Momentum = [tex]\frac{(x + 2)^{2}}{x - 3}[/tex] or [tex]\frac{x^{2} + 4x + 4}{x - 3}[/tex]
Caleb's mother said they will drive 1 /4 hour to get there, and they will drive another 1 /4 hour back. Once they arrive, Caleb's brother will be playing in 3 soccer games. Each soccer game takes 3 4 hour, and there is no break in between games. Calculate how long will they be away from home.
Answer: They are away for [tex]2\dfrac{3}{4}\ hours[/tex] from home.
Step-by-step explanation:
Since we have given that
Time to get there = [tex]\dfrac{1}{4}[/tex]
Time to get back = [tex]\dfrac{1}{4}[/tex]
Number of soccer games = 3
Time for each soccer games = [tex]\dfrac{3}{4}[/tex]
So, total time for 3 games would be
[tex]3\times \dfrac{3}{4}=\dfrac{9}{4}[/tex]
So, the time for which they will be away from home is given by
[tex]\dfrac{1}{4}+\dfrac{9}{4}+\dfrac{1}4}=\dfrac{1+9+1}{4}=\dfrac{11}{4}=2\dfrac{3}{4}\ hours[/tex]
Hence, they are away for [tex]2\dfrac{3}{4}\ hours[/tex] from home.
Caleb and his family will be away from home for a total of 2 3/4 hours.
To find out how long Caleb and his family will be away from home, we need to calculate the total time for the trip to and from the soccer games, as well as the duration of the games.
Driving to the location: 1/4 hourDriving back home: 1/4 hourEach soccer game lasts: 3/4 hour. Since there are 3 games, the total time spent in games is: 3 games × 3/4 hour/game = 2 1/4 hours.Adding it all together:
Total driving time: 1/4 hour + 1/4 hour = 1/2 hour
Total game time: 2 1/4 hours
Total time away from home: 1/2 hour + 2 1/4 hours = 2 3/4 hours
Therefore, Caleb and his family will be away from home for a total of 2 3/4 hours.
Solve for xxx. Your answer must be simplified. -30<\dfrac x{-4}−30< −4 x minus, 30, is less than, start fraction, x, divided by, minus, 4, end fraction
Answer:
[tex]x<120[/tex]
Step-by-step explanation:
We have been given an inequality [tex]-30<\frac{x}{-4}[/tex]. We are asked to solve the given inequality.
To solve for x, we will multiply both sides of inequality by negative 4. When we multiply or divide both sides of an inequality, the inequality sign reverses.
[tex]-30\cdot (-4)>\frac{x}{-4}\cdot (-4)[/tex]
[tex]120>x[/tex]
This means that 120 is greater than x or x is less than 120.
[tex]x<120[/tex]
Therefore, our required inequality would be [tex]x<120[/tex].
Final answer:
The solution to the inequality -30 < x / -4 is found by multiplying both sides by -4, which reverses the inequality sign, resulting in the solution x < 120. This demonstrates the manipulation of inequalities, particularly when involving negative multipliers.
Explanation:
The correct interpretation of this question is solving the inequality -30 < x / -4. To solve this inequality, we firstly multiply both sides by -4, remembering that multiplying or dividing by a negative number reverses the inequality sign. Thus, the inequality becomes 120 > x, which means x must be less than 120 for the inequality to hold true.
Therefore, the solution to the given inequality is x < 120. This highlights the importance of carefully handling inequalities, especially when multiplying or dividing by negative numbers, as it requires reversing the inequality sign to maintain the accurate relationship between both sides.
An ABA standard basketball can have a diameter of up to 25cm how much space is there for air inside a standard basketball
Answer:
[tex]8181.23 \ cm^3[/tex]
Step-by-step explanation:
-A standard basketball has a spherical shape.
-Given the ball has a diameter of 25cm.
-The space available for air is equivalent to the ball's volume and is calculated as:
[tex]V=\frac{4}{3}\pi r^3, D=25\\\\=\frac{4}{3}\pi (D/2)^3\\\\\frac{4}{3}\pi (25/2)^3\\\\=8181.23\ cm^3[/tex]
Hence, the space available for air is [tex]8181.23 \ cm^3[/tex]