The probability that exactly one customer dines on the first floor is:
0.26337
Step-by-step explanation:We need to use the binomial theorem to find the probability.
The probability of k success in n experiments is given by:
[tex]P(X=k)=n_C_k\cdot p^k\cdot (1-p)^{n-k}[/tex]
where p is the probability of success.
Here p=1/3
( It represents the probability of choosing first floor)
k=1 ( since only one customer has to chose first floor)
n=6 since there are a total of 6 customers.
This means that:
[tex]P(X=1)=6_C_1\times (\dfrac{1}{3})^1\times (1-\dfrac{1}{3})^{6-1}\\\\\\P(X=1)=6\times (\dfrac{1}{3})\times (\dfrac{2}{3})^5\\\\\\P(X=1)=0.26337[/tex]
Using the binomial distribution, it is found that there is a 0.2634 = 26.34% probability that exactly one customer dines on the first floor.
----------------
For each customer, there are only two possible outcomes, either they dine on the first floor, or they do not. The probability of a customer dining on the first floor is independent of any other customer, which means that the binomial probability distribution is used to solve this question.
----------------
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
And p is the probability of a success on a single trial.
----------------
Six customers, thus [tex]n = 6[/tex].They are equally as likely to dine on any of the three floors, thus [tex]p = \frac{1}{3} = 0.3333[/tex].----------------
The probability that exactly one customer dines on the first floor is P(X = 1), thus:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 1) = C_{6,1}.(0.3333)^{1}.(0.6667)^{5} = 0.2634[/tex]
0.2634 = 26.34% probability that exactly one customer dines on the first floor.
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100 frogs are released into a parkland lake. 80 % are expected to be green and the rest yellow. What is the number of yellow frogs that would be expected?
Answer:
20
Step-by-step explanation:
Well it is quite simple.You can find the 20% of 100 by multiplying 20 with 100 (wich means 20 yellow frogs,in this case within 100 frogs) and then diviting it with 100 (so it can be expressed as a
presentage that is based to 100).If you have any questions don't hesitate to contact me.
Yours sincerely,
Manos
A rectangular swimming pool measures 14 feet by 30 feet. The pool is
surrounded on all four sides by a path that is 3 feet wide. If the cost to
resurface the path is $2 per square foot, what is the total cost of
resurfacing the path?
I know that there is supposed to be some kind of second rectangle needed to find the answer(???), but I have no idea how to find it?? I asked my professor but he was really evasive and didn't give a solid answer.
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.
PLEASE HELP ME
Tony bought a desktop computer and a laptop computer. Before finance charges, the laptop cost $450 more than the desktop. He paid for the computers using two different financing plans. For the desktop the interest rate was 9% per year, and for the laptop it was 6 % per year. The total finance charges for one year were $300
. How much did each computer cost before finance charges?
Answer:
Laptop: $2,270
Desktop: $1,820
Step-by-step explanation:
Let L identify the laptop price and D the desktop price.
We can first say:
L = D + 450 ( the laptop cost $450 more than the desktop)
Then we can say:
0.09 D + 0.06 L = 300 (The total finance charges for one year were $300)
Then we substitute L by its value from first equation into the second equation:
0.09 D + 0.06 (D + 450) = 300
0.09 D + 0.06 D + 27 = 300
0.15D = 273 (removed 27 on both sides, and simplified left side)
D = 1,820
The cost of the desktop was $1,820
The cost of the laptop was $2,270 (price of desktop + $450)
Final answer:
By setting up equations based on the given finance charges and interest rates, we find that before finance charges, the desktop computer cost $1820 and the laptop cost $2270.
Explanation:
The student's question asks to determine the cost of each computer before finance charges. Let's denote the cost of the desktop computer as D and the cost of the laptop as L. From the information provided, we know that L = D + $450. The total finance charges for the desktop at 9% per year and for the laptop at 6% per year amount to $300. Hence, the equation for the finance charge can be written as 0.09D + 0.06L = $300. Substituting the expression for L from the first equation into the second, we get 0.09D + 0.06(D + $450) = $300, which simplifies to 0.09D + 0.06D + $27 = $300. Adding the D terms together, we get 0.15D + $27 = $300. Subtracting $27 from both sides, we obtain 0.15D = $273. Dividing both sides by 0.15, the cost of the desktop computer is found to be D = $1820. To find the cost of the laptop, we use the first equation: L = $1820 + $450 = $2270.
In conclusion, before finance charges, the desktop computer cost $1820 and the laptop $2270.
Margaret purchased a new bar of soap. Three days after she originally used the soap, she was curious how much soap per day she was using. She decided to weigh her soap and found that the bar was 103 grams. Four days later she re-measured the same bar of soap and recorded a weight of 80 grams. Assuming that Margaret uses the same amount of soap daily (and that she used the soap daily), write an equation that shows the amount of soap remaining after d days of use.
Answer:
The equation is:
[tex]y = 103- 23d[/tex]
Step-by-step explanation:
The initial amount was 103 grams.
After one day of use the remaining amount of soap was 80 grams.
So the amount of time he spent in one day was:
[tex]103-80 = 23[/tex]
Each day margaret spends 23 grams of soap.
if d represents the number of days elapsed then, the amount of soap "y" that Margaret spends after days is:
[tex]y = 103- 23d[/tex]
Margaret uses 5.75 grams of soap each day. The equation that shows the amount of soap remaining after d days of use is: S = 103 - 5.75d.
Explanation:Based on the information provided, we can find the rate of soap loss, measured in grams per day. Initially, Margaret's soap weighed 103 grams and 4 days later, it weighed 80 grams so we know that a total of 23 grams of soap was used over this 4-day period.
Therefore, Margaret is using (103-80) / 4 = 23 / 4 = 5.75 grams of soap each day. Given this daily usage rate, we can say that after d days, the amount of soap remaining can be calculated by subtracting the total soap used from the initial weight. So, our equation will be: S = 103 - 5.75d, where S represents the soap remaining and d represents the number of days.
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According to a certain central bank from 2000 to 2016 the average price of a new home in a certain region increased by 62 % to $470 thousand. What was the average price of a new home in 2000h The average price of a new home in 2000 was $ (Do not round until the final answer. Then round to the nearest thousand as needed)
Answer: $ 290 thousand
Step-by-step explanation:
Given : According to a certain central bank from 2000 to 2016 the average price of a new home in a certain region increased by 62 % to $470 thousand.
Let X be the the average price of a new home in 2000 .
Then , the 62 % increase in price is given by :-
[tex]x+0.62(x)=x(1+0.62)=1.62x[/tex]
Since , the the average price of the home in 2016 = $470 thosand
[tex]1.62x=470\\\\\Rightarrow\ x=\dfrac{470}{1.62}=290.12345679\approx290[/tex]
Hence, the average price of a new home in 2000 = $ 290 thousand .
The average price of a new house in 2000 is approximately 290,124 dollars.
What is the percentage?The amount of something is expressed as if it is a part of the total which is a hundred. The ratio can be expressed as a fraction of 100. The word percent means per 100. It is represented by the symbol ‘%’.
According to a certain central bank from 2000 to 2016 the average price of a new home in a certain region increased by 62 % to $470 thousand.
Let x be the average price of a new house in 2000. Then we have
[tex]\rm x = \dfrac{Present \ price}{1+Increased\ rate}\\\\\\x = \dfrac{470000}{1+0.62}\\\\\\x = \dfrac{470000}{1.62}\\\\\\x = 290123.4568 \approx 290124[/tex]
Thus, the average price of a new house in 2000 is 290,124 dollars.
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The diagram represents a pan balance each of the blocks marked x has the same value. The small blocks have a value of 1. What is the value of x if each side of the balance is the same
Answer:
1
Step-by-step explanation:
Blocks that have the same distance to the center cancel each other out.
The 4 rightmost and the 4 leftmost blocks cancel each other out.
In order to balance the remaining 4 1-blocks on the left side, the remaining right blocks must have the value 1.
Answer:
The correct option is B) 2.
Step-by-step explanation:
Consider the provided diagram.
There are 4 x blocks on the left side and 6 x blocks on the right side.
Also there are 4 small blocks have a value of 1.
Both the sides are balanced that means 4 x blocks + 4 small blocks equals to 6 x blocks.
4x + 4 = 6x
Subtract 4x from both the side.
4x + 4 - 4x = 6x - 4x
4 = 2x
Divide both the side by 2.
2 = x
Thus, the value of x is 2.
Hence, the correct option is B) 2.
A research group wants to determine whether the proportion of car accidents that were caused by drivers using cell phones has changed from the previous value of 13%. They obtained 10,000 auto accident reports and found that 14% were caused by drivers using cell phones. Find the test statistic.
The test statistic, z, is approximately 1.154. This value indicates a slight but potentially non-significant increase in the proportion of cell phone-related accidents compared to the previous value of 13%.
Explanation:Null and Alternative Hypotheses:
Null Hypothesis (H0): The proportion of accidents caused by cell phones has not changed, p = 0.13.
Alternative Hypothesis (Ha): The proportion has changed, p ≠ 0.13.
Test Statistic:
We can use the z-test for proportions to calculate the test statistic.
z = (Observed proportion - Expected proportion) / Standard Error
Observed proportion = 0.14 (14% from the sample)
Expected proportion = 0.13 (previous value)
Standard Error = sqrt(p * (1-p) / n) ≈ sqrt(0.13 * 0.87 / 10,000) ≈ 0.003
Calculation:
z = (0.14 - 0.13) / 0.003 ≈ 1.154
Therefore, the z-statistic is approximately 1.154.
Interpretation:
A z-score closer to 0 indicates no evidence against the null hypothesis (no change). Higher positive or negative values suggest increasing evidence for the alternative hypothesis (change). In this case, z = 1.154 is slightly positive, suggesting a potential but not conclusive increase in the proportion of cell phone-related accidents. Further analysis, such as p-value calculation, is needed to determine the statistical significance of this difference.
For the month of June in a certain city, 41% of the days are cloudy. Also in the month of June in the same city, 21% of the days are cloudy and rainy. What is the probability that a randomly selected day in June will be rainy if it is cloudy?
Answer:
0.5122 or 51.22%
Step-by-step explanation:
In a certain city, in June Probability of cloudy days = P(cloudy) = 0.41
Probability of cloudy and rainy = P(cloudy and rainy) = 0.21
Probability of rainy if we already know it is cloudy = [tex]\frac{\text{[P(cloud and rainy)]}}{[P(cloud)]}[/tex]
= [tex]\frac{0.21}{0.41}[/tex] = 0.512195122 ≈ 0.5122
Therefore, the probability that a randomly selected day in June will be rainy if it is cloudy is 0.5122 or 51.22%
The probability that a randomly selected day in June will be rainy if it is cloudy is approximately 51.22%.
To determine the probability that a randomly selected day in June will be rainy if it is cloudy, we can use conditional probability. The conditional probability formula is:
P(A|B) = P(A and B) / P(B)
Where,
P(A|B) is the probability that event A occurs given that B is true.P(A and B) is the probability that both A and B occur.P(B) is the probability that B occurs.Here, event A is 'rainy', and event B is 'cloudy'. Given data:
P(Cloudy) = 0.41P(Cloudy and Rainy) = 0.21To find the conditional probability P(Rainy | Cloudy), we apply the formula:
P(Rainy | Cloudy) = P(Cloudy and Rainy) / P(Cloudy) = 0.21 ÷0.41 ≈ 0.5122
So, the probability that a randomly selected day in June will be rainy if it is cloudy is approximately 0.5122, or 51.22%.
Use Newton's method with initial approximation x1 = −2 to find x2, the second approximation to the root of the equation x3 + x + 7 = 0. (Round your answer to four decimal places.)
Answer:
[tex]x_2 \approx -1.769[/tex]
Step-by-step explanation:
Let [tex]f(x)=x^3+x+7[/tex]
So [tex]f'(x)=3x^2+1[/tex]
[tex]x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}[/tex]
Let [tex]x_1=-2[/tex]
We are going to find [tex]x_2[/tex]
So we are evaluating [tex]-2-\frac{f(-2)}{f'(-2)}[/tex]
First step find f(-2)
Second step find f'(-2)
Third step plug in those values and apply PEMDAS!
[tex]f(-2)=(-2)^3+(-2)+7=-8-2+7=-10+7=-3[/tex]
[tex]f'(-2)=3(-2)^2+1=3(4)+1=12+1=13[/tex]
So
[tex]x_2=-2-\frac{-3}{13} \\\\ x_2=\frac{-26+3}{13} \\\\ x_2=\frac{-23}{13} \\\\ x_2 \approx -1.769[/tex]
The second approximation x2 using Newton's method for the equation x3 + x + 7 = 0 with an initial approximation of x1 = -2 is -2.2764.
Explanation:In order to find the second approximation x2 using
Newton's method
, we need to use the definition of Newton's method, which states that: x
n+1
= x
n
- f(x
n
)/f'(x
n
). Here, our function f(x) is x
3
+ x + 7. The derivative, f'(x), is 3x
2
+ 1. If our initial approximation, x1, is -2, we can substitute these values into our method to find x2. So, x2 = x1 - f(x1)/f'(x1) = -2 - ((-2)^3 + (-2) + 7) / (3*(-2)^2 + 1) = -2.2764 (rounded to four decimal places).
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Jane is saving her money in order to purchase a new racing bike. She initially saves $3 and plans to double the amount she saves each month. The bike Jane wants is $1,536 at the local bike shop.
Which equation represents this situation, and after how many months, t, will Jane have enough money to purchase the bike
Answer:
The equation is:
[tex]3(2) ^ t=1,536[/tex]
After [tex]t=9\ months[/tex]
Step-by-step explanation:
This situation can be represented by an exponential growth equation of the form
[tex]y = a (b) ^ {t}[/tex]
Where a is the initial amount
b is the growth rate
t is the time in months
In this case the initial amount is $ 3. Then [tex]a=3[/tex]
if she initially saves $3 and plans to double the amount she saves each month then
[tex]b=2[/tex]
The bike Jane wants is $1,536 at the local bike shop.
Then [tex]y=1,536[/tex]
The equation is:
[tex]3(2) ^ t=1,536[/tex]
Now we solve the equation for t
[tex]3(2) ^ t=1,536[/tex]
[tex](2) ^ t=\frac{1,536}{3}[/tex]
[tex](2) ^ t=512[/tex]
[tex]log_2(2) ^ t=log_2(512)[/tex]
[tex]t=log_2(512)[/tex]
[tex]t=9\ months[/tex]
Answer:
t=9 months. hope this helps
n 1895, the first Putting Green Championship was held. The winner’s prize money was $200. In 2016, the winner’s check was $1,480,000. a. What was the percentage increase per year in the winner’s check over this period? (Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.) b. If the winner’s prize increases at the same rate, what will it be in 2043?
Answer:
7.64% per year$10,805,480 . . . . rounded to 7 significant figuresStep-by-step explanation:
Using 2016 as a reference (t=0), the exponential equation for winnings can be written as ...
w(t) = 1480000×(1480000/200)^(t/121)
where 1480000 is the winnings in the reference year, and the ratio 1480000/200 is the ratio of winnings increase over the 121 years from 1895 to 2016.
This can be approximated by ...
w(t) ≈ 1,480,000×1.07640850764^t
In this form, we can see that the annual percentage increase is ...
1.0764 -1 = 7.64%
__
Then the winner's check in 2043, 27 years after 2016, is predicted to be ...
w(27) = $1,480,000×(1.0764...)^27 ≈ $10,805,478.41 ≈ $10,805,000
The percentage increase per year in the winner's prize money over the period is 6065.57%. The winner's prize money in 2043 would be approximately $15,190,712.55.
Explanation:To calculate the percentage increase per year, we need to find the average annual growth rate over the time period. First, we calculate the total percentage increase by taking the difference between the final and initial values, divided by the initial value.
In this case, it is (($1,480,000 - $200) / $200) * 100 = 740,000%. Then, we divide this percentage by the number of years, which is 2016 - 1895 + 1 = 122. So the annual percentage increase is 740,000% / 122 = 6065.57%.
To calculate the winner's prize in 2043, we need to find the number of years from 2016 to 2043, which is 2043 - 2016 = 27.
Then, we use the compound interest formula to calculate the future value: $1,480,000 * (1 + (6065.57% / 100))^27 = $15,190,712.55. So the winner's prize in 2043 would be approximately $15,190,712.55.
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A rectangular bird sanctuary is being created with one side along a straight riverbank. The remaining three sides are to be enclosed with a protective fence. If there are 28 km of fence available, find the dimension of the rectangle to maximize the area of the sanctuary.
Answer:
The rectangle is 7 km by 14 km. The 14 km dimension is parallel to the river.
Step-by-step explanation:
Let x represent the length of fence parallel to the river. The remaining fence is divided into two equal pieces for the ends of the enclosure. Then (28 -x)/2 will be the length of the side of the rectangle perpendicular to the river.
The total area of the enclosure is the product of length and width:
Area = (x)(28-x)/2
This expression describes a parabola opening downward with zeros at x=0 and x=28. The vertex (maximum) is halfway between those zeros, so is at ...
x = (0 +28)/2 = 14
Area is maximized when the dimension parallel to the river is 14 km and the ends of the enclosure are 7 km.
To maximize the area of the sanctuary, set up an equation with the length of the riverbank side. Differentiate and solve for x to find the dimensions of the rectangle.
Explanation:To maximize the area of the sanctuary, we need to find the dimensions of the rectangle.
Let the length of the riverbank side be x km.
The remaining two sides of the rectangle will each be (28 - x/2) km, as the total fence length should be equal to x km along the riverbank and (28 - x/2) km for the other two sides.
The area of the rectangle is given by A = x * (28 - x/2). To maximize the area, we can differentiate A with respect to x, set it equal to 0, and solve for x.
Taking the derivative of A, we get dA/dx = 28 - 3x/2. Setting this equal to 0, we find 28 - 3x/2 = 0. Solving for x, we get x = 18.67 km.
Therefore, the dimensions of the rectangle to maximize the area of the sanctuary are approximately 18.67 km along the riverbank and (28 - 18.67/2) km for the other two sides.
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Suppose that you need to create a list of n values that have a specific known mean. Some of the n values can be freely selected. How many of the n values can be freely assigned before the remaining values are determined
Final answer:
You can assign up to (n-1) values freely in a list of n values that must have a specific known mean. The last value is then determined by ensuring the sum of all n values achieves the required total that reflects the known mean.
Explanation:
To construct a list of n values with a predetermined mean, you can think of the sum total that these n values should add up to, as the mean (let's call it μ) multiplied by the number of items in the list, n. If you want to freely assign a certain number of values, let's call it k, then these k values can be anything that respects the constraints of the data (like being positive if you're measuring something that can't be negative). Once you have assigned these k values, the sum of the remaining (n-k) values is determined because it must make up the difference needed to reach the predetermined total sum that corresponds to the known mean. Therefore, you can freely assign up to (n-1) values and the last value will be determined by the mean constraint.
Does the function satisfy the hypotheses of the Mean Value Theorem on the given interval? f(x) = 4x^2 + 3x + 4, [−1, 1]
No, f is continuous on [−1, 1] but not differentiable on (−1, 1).
No, f is not continuous on [−1, 1].
Yes, f is continuous on [−1, 1] and differentiable on (−1, 1) since polynomials are continuous and differentiable on .
There is not enough information to verify if this function satisfies the Mean Value Theorem.
Yes, it does not matter if f is continuous or differentiable; every function satisfies the Mean Value Theorem.
Answer:14
Step-by-step explanation:
Use this list of Basic Taylor Series to find the Taylor Series for f(x) = 2x 1+x2 based at 0. Give your answer using summation notation and give the largest open interval on which the series converges. (If you need to enter ∞ , use the ∞ button in CalcPad or type "infinity" in all lower-case.) The Taylor series for f(x)= 2x 1+x2 is: ∞ k=0 The Taylor series converges to f(x) for |x|< . Use the Taylor series you just found for f(x)= 2x 1+x2 to find the Taylor series for ln(1+x2) based at 0. Give your answer using summation notation and give the largest open interval on which the series converges. The Taylor series for ln(1+x2) is: ∞ k=0 The Taylor series converges to ln(1+x2) for |x|< .
Recall that for [tex]|x|<1[/tex], we have
[tex]\displaystyle\frac1{1-x}=\sum_{n=0}^\infty x^n[/tex]
Then for [tex]|-x^2|<1[/tex], or [tex]|x|<1[/tex], we have
[tex]\displaystyle\frac1{1+x^2}=\frac1{1-(-x^2)}=\sum_{n=0}^\infty(-x^2)^n=\sum_{n=0}^\infty(-1)^nx^{2n}[/tex]
Multiply this series by [tex]2x[/tex] to get the Taylor series for [tex]f(x)[/tex]:
[tex]f(x)=\dfrac{2x}{1+x^2}=\displaystyle2\sum_{n=0}^\infty(-1)^nx^{2n+1}[/tex]
Notice that
[tex]\dfrac{\mathrm d(\ln(1+x^2))}{\mathrm dx}=\dfrac{2x}{1+x^2}[/tex]
so to find the Taylor series for [tex]g(x)=\ln(1+x^2)[/tex], we integrate the Taylor series for [tex]f(x)[/tex]:
[tex]g(x)=\displaystyle\int f(x)\,\mathrm dx=C+2\sum_{n=0}^\infty\frac{(-1)^nx^{2n+2}}{2n+2}[/tex]
Since [tex]g(0)=\ln(1+0^2)=\ln1=0[/tex], it follows that [tex]C=0[/tex] and
[tex]g(x)=\displaystyle\sum_{n=0}^\infty\frac{(-1)^nx^{2n+2}}{n+1}[/tex]
which converges for [tex]|x|<1[/tex] as well.
Following are the calculation to the Taylor Series:
Geometric series:
[tex]\to \Sigma_{\infty}^{n=0} \ a+ar+ar^2+.....+[/tex] which converges to [tex]\frac{a}{1-r} \ \ for\ \ |r| < 1[/tex].
Remembering that:
[tex]\to \frac{2x}{1+x^2}=\frac{2x}{1-(-x^2)}[/tex] Taking [tex]a=2x \ \ \ \ \ \ \ \ \ r=-x^2\\\\[/tex]
Using the Taylor series:
[tex]\to \frac{2x}{1+x^2}= \Sigma_{\infity}^{n=0} \ 2x \times (-x^2)^{n} = \Sigma_{\infity}^{n=0} 2x \times (-1)^{n} \times (x^{2n})[/tex]
[tex]\to \frac{2x}{1+x^2}= \Sigma_{\infty}^{n=0} (-1)^{n} \times 2 \times x^{2n+1} = 2x -2x^3+2x^5+............+[/tex]
In the given scenario we will converge the [tex]|x^2| < 1 |x| < 1[/tex]. Now, realize:
[tex]\to \frac{2x}{1+x^2} \ dx = \In (1+x^2) \\\\[/tex]
Integrating the series for [tex]\frac{2x}{1+x^2}[/tex] :
[tex]\to \In( 1+x^2)=\int \Sigma_{\infty}^{n=0} \ (-1)^{n} \times 2 \times x^{2n+1}\ dx\\\\[/tex]
[tex]=\Sigma_{\infity}^{n=0} \frac{(-1)^{n} \times 2 \times x^{2n+2}}{ 2n+2}\\\\=\Sigma_{\infity}^{n=0} \frac{(-1)^{n} \times x^{2n+2}}{ n+1}\\\\= x^2 -\frac{x^4}{2}+\frac{x^6}{3}+............+[/tex]
Since integrating a number has no effect on its radius of converge, this series similarly converges for [tex]\bold{|x| < 1}[/tex].
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The graph of f(x)=|x| is reflected over the y-axis and horizontally compressed by a factor of 1/9. Write a formula for function g(x)
The reflection and the horizontal compressions are illustrations of transformations.
The formula for function g(x) is [tex]\mathbf{g(x) = 9x}[/tex]
The function is given as:
[tex]\mathbf{f(x) = |x|}[/tex]
The rule of reflection over the y-axis is:
[tex]\mathbf{(x,y) \to (-x,y)}[/tex]
So, we have:
[tex]\mathbf{f'(x) = |-x|}[/tex]
[tex]\mathbf{f'(x) = x}[/tex]
The rule of horizontal compression is:
[tex]\mathbf{(x,y) \to (\frac xb,y)}[/tex]
So, we have:
[tex]\mathbf{g(x) = \frac{x}{1/9}}[/tex]
[tex]\mathbf{g(x) = 9x}[/tex]
Hence, the formula for function g(x) is [tex]\mathbf{g(x) = 9x}[/tex]
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A reflection over the y-axis changes x to -x and a horizontal compression by a factor of 1/9 replaces x by 9x. Hence, the function g(x) reflecting these transformations is |-9x|.
Explanation:The original function is f(x) = |x|. When a function is reflected over the y-axis, it changes x to -x. Hence the function becomes f(-x) = |-x|. A compression by a factor of 1/9 in the horizontal direction is represented by replacing x by 9x, our function becomes f(9x) = |-9x|. So, the new function g(x) = f(-9x) = |-9x|.
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One common system for computing a grade point average (GPA) assigns 4 points to an A, 3 points to a B, 2 points to a C, 1 point to a D, and 0 points to an F. What is the GPA of a student who gets an A in a 33-credit course, a B in each of threethree 44-credit courses, a C in a 22-credit course, and a D in a 33-credit course?
Answer:
Step-by-step explanation:
If i am not mistaken I believe it is 2.75 lemme know if that is right!
What are the solutions of the equation x4 – 5x2 – 14 = 0? Use factoring to solve.
Someone help please!!!
Answer: x=√7
x=√2i
We'll use factoring by grouping to solve the equation. This method involves grouping the terms of the polynomial into two binomials, such that the product of the leading coefficients of the binomials is equal to the constant term, and the sum of the products of the remaining terms is equal to the middle term.Steps to solve:
1. Factor the expression:
(x²−7)(x²+2)=0. Create separate equations and solve:
x²−7=0
x²+2=0. Solve the first equation:
x²−7=0
x=±√7. Solve the second equation:
x²+2=0
x=±√2i
I need help with math
Answer:
Let's say that the mile that the cyclist going west is a.
And so the one heading North is (a+5)
a+(a+5)=25
a+a+5=25
2a=20
a=10
So the one heading West has traveled 10 mi.
geom help please will give brainliest
The Venn Diagram represents a group of children who swam (left circle) and built sandcastles (right circle) at the beach.
Match the symbol or description on the left with its corresponding value on the right. S stands for the event “Swam” and C stands for the event “built sandcastles.” Assume the numbers represent the entire universe.
1. P(S)
1.00
2. P(S or C, but not both)
0.84
3. P(C)
0.16
4. P(S ∪ C)
0.60
5. P(C, but not S)
0.76
6. P(S ∩ C)
0.40
Try this option:
1] P(S)=0.84;
2] P(S or C, but not both)=0.4;
3] P(C)=0.76;
4] P(S∪C)=0.6;
5] P(C, but not S)=0.16;
6] P(S∩C)=1.00.
Answer:
We are given with a Venn diagram.
In Venn Diagram,
S represent Swam
C represent Built Sandcastles.
n( S - (S∩C) ) = 6
n( C - (S∩C) ) = 4
n( S ∩ C ) = 15
To find: P(S) , P(C) , P(S or C, but not Both) = P((S∪C) - (S∩C)) , P( S ∪ C ) ,
P(S ∩ C) , P(C , but not S ) = P(C - (S∩C))
n(S) = n( S - (S∩C) ) + n(S∩C) = 6 + 15 = 21
n(C) = n( C - (S∩C) ) + n(S∩C) = 4 + 15 = 19
n(S∪C) = n( C - (S∩C) ) + n( S - (S∩C) ) + n(S∩C) = 6 + 4 + 15 = 25
Now, [tex]P(S)=\frac{21}{25}=0.84[/tex]
[tex]P(S\:or\:C,\:but\:not\:Both)=P((S\cup C)-(S\cap C))=\frac{10}{25}=0.40[/tex]
[tex]P(C)=\frac{19}{25}=0.76[/tex]
[tex]P(S\cup C)=\frac{25}{25}=1.00[/tex]
[tex]P(C\:,\:but\:not\:S)=P(C - (S\cap C))=\frac{4}{25}=0.16[/tex]
[tex]P(S\cap C)=\frac{15}{25}=0.60[/tex]
Therefore, Match the answers as above.
which simplifys to a+ b?
a-(1-b)+1
-(1-a)-b+1
(a-1)-(b-1)
a-(-b-1)+1
Answer:
see below
Step-by-step explanation:
Use the distributive property to eliminate parentheses. Remember that the product (-1)(-1) is 1.
a-(1-b)+1 = a -1 +b +1 = a + b . . . . this one
__
-(1-a)-b+1 = -1 +a -b +1 = a - b
__
(a-1)-(b-1) = a - 1 - b + 1 = a - b
__
a-(-b-1)+1 = a +b +1 +1 = a + b + 2
Answer:
idiidhdmfnrbbbbbbrh
Step-by-step explanation:
yes
Find parametric equations for the line. (Enter your answers as a comma-separated list of equations. Let x, y, and z be functions of t.) The line in the direction of the vector 5 i + 5 j − 6k and through the point (−4, 4, −2).
Answer:
x=5t-4 , y=5t+4 , z=-6t-2
Step-by-step explanation:
So we are going to use (-4,4,-2) as an initial point, p.
The direction vector is v=5i+5j-6k or <5,5,-6>.
The vector equation is r=vt+p.
That means we have r=<5,5,-6>t + <-4,4,-2>.
So the parametric equations are
x=5t-4
y=5t+4
z=-6t-2
The parametric equations are:
x = -4 + 5t
y = 4 + 5t
z = -2 - 6t
The given direction vector is:
[tex]\bar{V} = 5i + 5j - 6k[/tex]
The direction vector can also be written as:
[tex]\bar{V} = <a, b, c> = <5, 5, -6>[/tex]
The point X₀ = (x₀, y₀, z₀) = (-4, 4, -2)
The parametric equation is of the form:
[tex]X = X_{0} + \bar{V}t[/tex]
This is:
[tex]\left[\begin{array}{ccc}x\\y\\z\end{array}\right] = \left[\begin{array}{ccc}x_0\\y_0\\z_0\end{array}\right] + \left[\begin{array}{ccc}a\\b\\c\end{array}\right]t[/tex]
[tex]\left[\begin{array}{ccc}x\\y\\z\end{array}\right] = \left[\begin{array}{ccc}-4\\4\\-2\end{array}\right] + \left[\begin{array}{ccc}5\\5\\-6\end{array}\right]t[/tex]
The parametric equations are therefore:
x = -4 + 5t
y = 4 + 5t
z = -2 - 6t
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13. Justin generously lends $500,000 to his friend, Jamie. However, being a maleficent businessman, he charges Jamie interest at 8.6 % per annum compounded quarterly. How much will Jamie owe Justin after 25 years? What is the accumulated interest? [3 marks]
Answer:
Jamie will owe $ 41,95,725.83 ( approx ),
Accumulated interest is $ 36,95,725.83
Step-by-step explanation:
Since, the amount formula in compound interest is,
[tex]A=P(1+\frac{r}{n})^{nt}[/tex]
Where, P is the principal amount,
r is the annual rate of interest,
n is the compounding periods in a year,
t is the time in years,
Here, P = $ 500,000,
r = 8.6 %=0.086,
n = 4,
t = 25 years,
By substituting the values,
[tex]A=500000(1+\frac{0.086}{4})^{100}[/tex]
[tex]=500000(1+0.0215)^{100}[/tex]
[tex]=500000(1.0215)^{100}[/tex]
[tex]=4195725.82746[/tex]
[tex]\approx 4195725.83[/tex]
Also, the accumulated interest = A-P = 4195725.83 - 500000 = $ 3695725.83
Find the volume of a solid enclosed by the paraboloid z = x2 +y2 and a plane z = 9
The plane [tex]z=9[/tex] lies above the paraboloid [tex]z=x^2+y^2[/tex], so the volume of the bounded region [tex]R[/tex] is given by
[tex]\displaystyle\iiint_R\mathrm dV=\int_{-3}^3\int_{-\sqrt{9-x^2}}^{\sqrt{9-x^2}}\int_{x^2+y^2}^9\mathrm dz\,\mathrm dy\,\mathrm dx[/tex]
Convert to cylindrical coordinates, setting
[tex]\begin{cases}x=r\cos\theta\\y=r\sin\theta\\z=z\end{cases}\implies\mathrm dx\,\mathm dy\,\mathrm dz=r\,\mathrm dr\,\mathrm d\theta\,\mathrm dz[/tex]
and the integral is equivalent to
[tex]\displaystyle\int_0^{2\pi}\int_0^3\int_{r^2}^9r\,\mathrm dz\,\mathrm dr\,\mathrm d\theta=2\pi\int_0^3(9r-r^3)\,\mathrm dr=\boxed{\frac{81\pi}2}[/tex]
The volume of the solid enclosed by the paraboloid z = x² + y² and the plane z = 9 is found by using double integrals in polar coordinates. The volume is calculated as 81π cubic units.
Explanation:To find the volume of a solid enclosed by the paraboloid z = x² + y² and the plane z = 9, you have to use the method of double integrals in polar coordinates. The cone extends from z = 0 at its apex to z = 9 at the top, which is given by the plane. Hence, we can imagine this region as a bunch of thin disks or pancakes that lie above circles in the xy-plane and pile up to form the parcel of the parabolic solid under the plane z = 9.
In this case, we have to integrate over the region R, which is a disk of radius 3 (it's the projection on the xy-plane under the plane z = 9), with the height of a 'thin disk' as z = x² + y² = r² (in polar coordinates). Therefore, the volume V can be given as:
V = ∫∫R(z*r*dr*dθ) = ∫02π∫03(r²*r*dr*dθ) = 2π* [03 0.25r⁴] = 2π*(40.5-0) = 81π cubic units.
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what is the solution of the associated homogeneous equation y'' + y = 0
Answer: [tex]y(t)=Acos(t)+Bsin(t)[/tex]
Step-by-step explanation:
To find the solution of a given differential equation ay''+by'+cy=0, a≠0, you have to consider the quadratic polynomial ax²+bx+c=0, called the characteristic polynomial.
Using the quadratic formula, this polynomial will always have one or two roots, for example r and s. The general solution of the differential equation is:
[tex]y(t)= Ae^{rt}+Be^{st}[/tex] , if the roots r and s are real numbers and r≠s.
[tex]y(t)= A e^{rt}+B*t*e^{rt}[/tex] , if r=s is real.
[tex]y(t)=Acos(\beta t)e^{\alpha t} +Bsin(\beta t)e^{\alpha t}[/tex] , if the roots r and s are complex numbers α+βi and α−βi
.
In this case, the characteristic polynomial is:
[tex]x^{2} +1=0\\x^{2} =-1\\x1=i; x2=-i[/tex]
Since the roots are complex numbers, with α=0 and β=1, then the answer is: [tex]y(t)=Acos(t)+Bsin(t)[/tex]
Write an equation for the given function given the period, phase shift, and vertical shift.
cotangent function, period = π, phase shift = -1/3 π, vertical shift = 2.
ANSWER
[tex]y = \cot(x - \frac{\pi}{3} ) + 2[/tex]
EXPLANATION
The cotangent function that is fully transformed is of the form
[tex]y =a \cot(bx + c) + d[/tex]
where 'a' is the amplitude.
[tex] \frac{\pi}{b} = \pi[/tex]
is the period.
This implies that b=1
The phase shift is
[tex] \frac{c}{b} = - \frac{\pi}{3} [/tex]
Substitute b=1 to get;
[tex]c = - \frac{\pi}{3} [/tex]
and d=2 is the vertical shift.
We choose a=1 to get the required function as
[tex]y = \cot(x - \frac{\pi}{3} ) + 2[/tex]
if i have 18 days to complete assignments and i have 44 assignments to do, how many assignments do i have to do a day?
For 11 days, you'd do 3 assignments. That'll knock off 33 assignments. Then, for 5 days, you'll 2 assignments, which will leave you with 2 assignments. Then for one day, you'll only have to do 1 assignment. The last day you are free!!! :)
Consider the function f(x)=-3x^2 +7x -k. [3 Marks] a) For what values of k will the function have no zeros? b) For what values of k will the function have one zero? c) For what values of k will the function have two zeros?
Answer: a) k >4.08
b) k = 4.08
c) k<4.08
Step-by-step explanation:
Since we have given that
[tex]f(x)=-3x^2+7x-k[/tex]
a) For what values of k will the function have no zeros?
It mean it has no real zeroes i.e. Discriminant < 0
As we know that
[tex]D=b^2-4ac[/tex]
Here, a =-3
b = 7
c = -k
So, it becomes,
[tex]D<0\\\\b^2-4ac<0\\\\7^2-4\times -3\times -k<0\\\\49-12k<0\\\\-12k<-49\\\\k>\dfrac{49}{12}\\\\k>4.08[/tex]
b) For what values of k will the function have one zero?
It means it has one real root i.e equal roots.
So, in this case, D = 0
So, it becomes,
[tex]D=b^2-4ac=0\\\\D=7^2-4\times -3\times -k=0\\\\49-12k=0\\\\49=12k\\\\k=\dfrac{49}{12}\\\\k=4.08[/tex]
c) For what values of k will the function have two zeros?
It means it has two real roots.
In this case, D>0
So, it becomes,
[tex]D=7^2-4\times -3\times -k>0\\\\49-12k>0\\\\-12k>-49\\\\12k<49\\\\k<4.08[/tex]
Hence, a) k >4.08
b) k = 4.08
c) k<4.08
How do I simply this expression (quadratic formula basis) on a TI-84 or normal calculator?
Answer:
[tex]1000\pm 100\sqrt{55}[/tex]
Step-by-step explanation:
A TI-84 or "normal" calculator is designed to evaluate expressions numerically. It can tell you the numerical value of this expression is the set of values
{1741.619849, 258.3801513}
but it cannot simplify the expression.
This expression can be simplified by evaluating the fraction and removing double factors from under the radical:
[tex]\dfrac{2000\pm\sqrt{2200000}}{2}=\dfrac{2000}{2}\pm\sqrt{\dfrac{2200000}{2^2}}=1000\pm\sqrt{550000}\\\\=1000\pm\sqrt{100^2\cdot 55}=1000\pm 100\sqrt{55}[/tex]
From a group of 12 students, we want to select a random sample of 4 students to serve on a university committee. How many different random samples of 4 students can be selected?
Answer:
Step-by-step explanation:
3