Express cos K as a fraction in simplest terms.
M
√51
12
K

If Charlotte will take out the garbage for 15 weeks,  Charlotte will earn 327.67 dollars on the 15th week.

To find how much Charlotte will earn on the 15th week, we can use the formula for the sum of a geometric series:

Sₙ = a(1 - rⁿ) / (1 - r)

where Sₙ is the sum of the first n terms of the series, a is the first term, r is the common ratio, and n is the number of terms.

In this case, a = 1 cent, r = 2 (since each week Charlotte earns twice as much as she did the week before), and n = 15. Substituting these values into the formula gives:

S₁₅ = 1(1 - 2¹⁵) / (1 - 2)

S₁₅ = (1 - 32768) / (-1)

S₁₅ = 32767 cents

Therefore, Charlotte will earn 327.67 dollars on the 15th week.

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We can see here that the vertices will be:

The vertex, in geometry, is the intersection of two or more lines, curves, or edges. It can also refer to the vertex of a parabola, which is where a function reaches its highest or lowest value.

We can see here that the equation of the hyperbola is seen in standard form. It is known that the center of the hyperbola is at (h, k) is (2, 1).

The distance between the center and vertices = a

where a² = coefficient of the positive term

So we see that  a² = 16

a = 4.

Also, the distance between the center and co-vertices = b

where b² = 4

b = 2.

Thus,

Vertex 1 = (2 + 4, 1) = (6, 1)

Vertex 2 = (2 - 4, 1) = (-2, 1).

Therefore, the vertices are:

(6, 1) and (-2, 1).

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Answer:

v would be 28 degrees

Step-by-step explanation:

You do the total angle take away by 43.

71 - 43 = 28

a) To calculate the probability that the sample proportion of defective cups from Supplier A is less than 10%, we need to find the probability that the sample proportion is less than 0.10. Thus, we need to find P(p < 0.10).

We can use the central limit theorem to approximate the distribution of the sample proportion as a normal distribution, with mean μ = 0.08 and standard deviation [tex]σ = \sqrt{0.08 (\frac{1-0.08)}{200} )}= 0.024[/tex]. Then, we can standardize the distribution and use a standard normal table or calculator to find the probability:

[tex]P (p < 0.10)=P(\frac{p-u}{σ} < \frac{0.10-0.08}{0.024} = P(z < 0.83)=0.7977[/tex]

Therefore, the probability that the sample proportion of defective cups from Supplier A is less than 10% is approximately 0.7977.

b) To calculate the probability that the sample proportion of defective cups from Supplier A is at least 5%, we need to find the probability that the sample proportion is greater than or equal to 0.05. Thus, we need to find P(p≥ 0.05).

Using the same approach as in part (a), we can find that                                                             P(p < 0.05)=-0.0207. Therefore, P(p ≥ 0.05) = 1 - P(p< 0.05) =0.9793.

Therefore, the probability that the sample proportion of defective cups from Supplier A is at least 5% is approximately 0.9793.

c) To calculate the probability that the sample proportion of defective cups from Supplier A is between 5% and 10%, we need to find the probability that 0.05 ≤ p < 0.10. We can use the same approach as in part (a) to find that P(p < 0.05) = 0.0207 and P(p < 0.10) = 0.7977. Therefore, P(0.05 ≤ p < 0.10) = P(p < 0.10) - P(p < 0.05) = 0.7770.

Therefore, the probability that the sample proportion of defective cups from Supplier A is between 5% and 10% is approximately 0.7770.

d) To calculate the probability that the sample proportion of defective cups from Supplier A is exactly 8%, we need to find P(p = 0.08). Since the sample proportion is a discrete random variable, we can use the binomial distribution to find the probability:

[tex]P(p=0.08)=(200 choose 16) (0.080)^{16} (1-0.08)^{184} = 0.1567[/tex]

Therefore, the probability that the sample proportion of defective cups from Supplier A is exactly 8% is approximately 0.1567.

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We can expect about 84 watermelons to weigh less than 17.36 pounds.

We have,

To answer this question, we need to use the concept of standard normal distribution.

First, we need to calculate the z-score of 17.36 using the formula:
z = (x - μ) / σ
where x is the weight we're interested in, μ is the mean weight, and σ is the standard deviation. Substituting the values given in the question, we get:
z = (17.36 - 20) / 2.9
z = -0.9655

Now, we can look up the area under the standard normal curve to the left of z = -0.9655 using a z-table or a calculator. The result is 0.1675.

This means that about 16.75% of the watermelons will weigh less than 17.36 pounds.

To find the actual number of watermelons, we can multiply this percentage by the total number of watermelons produced:
500 x 0.1675 = 83.75

Therefore,

We can expect about 84 watermelons to weigh less than 17.36 pounds.

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The probability of exactly one successes in five trials  is 0.20

From the question, we have the following parameters that can be used in our computation:

The probability is calculated as

P(x) = nCx * p^x * (1 - p)^(n -x)

Where

n = 5

p = 5%

x = 1

Substitute the known values in the above equation, so, we have the following representation

P(1) = 5C1 * (5%)^1 * (1 - 5%)^(5 -1)

Evaluate

P(1) = 0.20

HEnce, the probability value is 0.20

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Answer:

To find out how many minutes Valeria would practice in 4 weeks, we need to first find out how many minutes she practices per week.

Divide the total number of minutes she practices by the number of weeks she practices:

910 minutes ÷ 5 weeks = 182 minutes per week

Valeria practices 182 minutes per week.

To find out how many minutes she would practice in 4 weeks, we can multiply the minutes per week by the number of weeks:

182 minutes/week x 4 weeks = 728 minutes in 4 weeks

Valeria would practice 728 minutes in 4 weeks.

A. The domain is N and the image is the set of all natural numbers that are products of two natural numbers

B. The image is all values of y in Q such that (1 - 2x)/3 is defined.

C. we note that y² + 1 is always odd, so the image is the set of all odd natural numbers greater than or equal to 2.

What is function?

In mathematics, a function is a relationship between two sets of elements, called the domain and the range, such that each element in the domain is associated with a unique element in the range.

(a) R is a function because for each x in N, there exists a unique y in N such that xy. The domain is N and the image is the set of all natural numbers that are products of two natural numbers.

(b) R is a function because for each x in Q, there exists a unique y in Q such that 2x+3y=1. To find the domain, we solve for x in terms of y: 2x = 1 - 3y, x = (1 - 3y)/2. The domain is all values of y in Q such that (1 - 3y)/2 is defined. Simplifying, we get y ≠ 1/3. Therefore, the domain is Q - {1/3}. To find the image, we solve for y in terms of x: 3y = 1 - 2x, y = (1 - 2x)/3. The image is all values of y in Q such that (1 - 2x)/3 is defined. Therefore, the image is Q.

(c) R is a function because for each x in N, there exists a unique y in N such that y²+1=x. To find the domain, we solve for x in terms of y: y² = x - 1, y = ±√(x - 1). Since we are given that x and y are natural numbers, the domain is the set of all natural numbers greater than or equal to 2. To find the image, we note that y² + 1 is always odd, so the image is the set of all odd natural numbers greater than or equal to 2.

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Answer:

The kinetic energy (KE) of an object with mass (m) moving at a velocity (v) can be calculated using the formula:

KE = 1/2 * m * v^2

Substituting the given values:

KE = 1/2 * 1200 kg * (8.33 m/s)^2

KE = 41,147.5 J

Therefore, the kinetic energy of the object is 41,147.5 Joules.

Step-by-step explanation:

If a candy bar cost 95 cents, then I'll cost 3.8$ to buy the four candy's

To Find How much will it cost to buy 4 candy bars which comes in price of 95 cents per candy bar so therefore calculation will be:

One candy bar cost  = 95 cent

We will use multiplication to multiply the cost of each candy.

4 candy bar costs  = 4 x  price of one candy

4 candy bar costs  = 4 x  95 cents

4 candy bar costs  = 380 cents

After that we get:

100 cents = 1 $

Now we will divide:

4 candy bar costs  = 380/100$

4 candy bar costs  = 3.8 $

Therefore, If a candy bar is 95 cents, it will cost me 3.8$ to purchase the four candies together.

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The value of expression is,

⇒ x ÷ 12

We have to given that;

The algebraic expression is,

⇒ x divided by 12

Hence, We can formulate;

The value of correct expression is,

⇒ x ÷ 12

⇒ x / 12

Thus, The value of expression is,

⇒ x ÷ 12

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Answer:

First, we need to calculate the total exemptions for Kareem, his wife, and their child:

Total exemptions = 3 x 4,050 = 12,150

Next, we subtract the standard deduction and exemptions from their adjusted gross income to find their taxable income:

Taxable income = 92,600 - 12,600 - 12,150 = 67,850

Therefore, the correct answer is (B) 67,850.

Step-by-step explanation:

Branliest please

For Daniel's four-day business trip, the total fixed costs for the car rental from car rental agency is equals the $153.2.

We have, Daniel plans to rent a car for an upcoming four-day business trip.

Flat fee charges for rent a car from car rental agency = $29 per day

Charges for driven = $0.12 per mile

Total distance travelled by him on first day = 140 miles

Cost of driven charges on first day = 140× 0.12 = $16.8

Total distance travelled by him on secon day = 15 miles

Cost of driven charges on first day = 15× 0.12 = $1.8

Total distance travelled by him on third day = 15 miles

Cost of driven charges on first day = 15× 0.12 = $1.8

Total distance travelled by him on fourth day = 140 miles

Cost of driven charges on first day = 140 × 0.12 = $16.8

Total cost of driven charges on four-day business trip = $16.8 + $16.8 + $1.8 + $1.8

= $37.2

Now, total fixed cost for rent a car are calculated by sum of driven charges and flat fee for rent = $37.2 + 4×$29

= $153.2

Hence required value is $153.2.

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Answer:

replace

y

with

0

and solve for

x

.

Step-by-step explanation:

x=-1/3,2

Arun is going to invest $7,700 and leave it in an account for 20 years. Assuming theinterest is compounded continuously, what interest rate, to the nearest hundredth ofa percent, would be required in order for Arun to endend up with $13,100?

So first you do 13,100 minus 7700 equals 5400. Now you have 5400 you want to divide it by 20 which equals 270. So over the course of 20 years it went up to 13100. Which means every year it had to go up by $270 but that’s not a percent so… we have to divide 13100 by 5400 which equals 2.43 (Hope this helps)

We first calculate D⁴:

D⁴ = |1⁴ 0 0 |

|0 2⁴ 0 |

|0 0 4⁴|

Substituting into the formula, we get:

A⁴ =

Question 1:

To find the eigenvalues and eigenvectors of matrix A, we solve the characteristic equation:

|A - λI| = 0

where I is the identity matrix and λ is the eigenvalue.

Substituting A, we get:

|1-λ 1 3 |

|1 5-λ 1 | = 0

|3 1 1-λ|

Expanding the determinant, we get:

(1-λ) [(5-λ)(1-λ) - 1] - (1)[(1)(1-λ) - (3)(1)] + (3)[(1)(1) - (5-λ)(3)] = 0

Simplifying, we get:

-λ³ + 7λ² - 14λ + 8 = 0

This equation can be factored as:

-(λ-1)(λ-2)(λ-4) = 0

Therefore, the eigenvalues of A are λ1 = 1, λ2 = 2, and λ3 = 4.

To find the eigenvectors, we solve the equation (A-λI)x = 0 for each eigenvalue.

For λ1 = 1, we get:

|0 1 3 | |x1| |0|

|1 4 1 | |x2| = |0|

|3 1 -0 | |x3| |0|

Simplifying, we get the system of equations:

x2 + 3x3 = 0

x1 + 4x2 + x3 = 0

3x1 + x2 = 0

Solving this system, we get:

x1 = -3x3

x2 = x3

x3 = x3

So, the eigenvector corresponding to λ1 = 1 is:

v1 = (-3, 1, 1)

Similarly, for λ2 = 2, we get:

v2 = (-1, 1, -1)

And for λ3 = 4, we get:

v3 = (1, 1, -3)

Therefore, the eigenvalues of A are 1, 2, and 4, and the corresponding eigenvectors are (-3, 1, 1), (-1, 1, -1), and (1, 1, -3).

Question 2:

To find the matrix P that transforms A to diagonal form, we need to find the eigenvectors of A and use them as columns of P. That is:

P = [v1 v2 v3]

where v1, v2, and v3 are the eigenvectors of A.

From Question 1, we have:

v1 = (-3, 1, 1)

v2 = (-1, 1, -1)

v3 = (1, 1, -3)

So, the matrix P is:

P = |-3 -1 1|

| 1 1 1|

| 1 -1 -3|

To calculate A⁴, we use the formula:

Aⁿ = PDⁿP⁻¹

where Dⁿ is the diagonal matrix with the eigenvalues raised to the nth power.

So, we first calculate D⁴:

D⁴ = |1⁴ 0 0 |

|0 2⁴ 0 |

|0 0 4⁴|

Substituting into the formula, we get:

A⁴ =

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To determine the value of "r" between age and salary, we would need to conduct a statistical analysis, such as a correlation coefficient calculation. Without this information, it is impossible to determine the direction or strength of the relationship between age and salary.

Similarly, without the results of a regression analysis, it is not possible to determine the value of R², which represents the proportion of variance in the dependent variable (salary) that can be explained by the independent variable (age). Once this value is known, we can use the rubric to determine the strength of the predictor.

However, based on the rubrics provided, if the correlation coefficient (r) is close to -1 or 1, the relationship between age and salary would be considered very strong, either negatively or positively correlated. If the coefficient is closer to 0, the correlation would be considered weak or somewhat weak.

Similarly, R² measures the proportion of variance in the dependent variable (salary) that is explained by the independent variable (age). A value of 1 would indicate a perfect predictor, while a value of 0 would indicate no relationship between the variables. Values between 0 and 1 would indicate varying degrees of predictive power.

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The actual length and width of the kitchen in the scale drawing is: 22.5 ft x 17.5 ft

A scale drawing is defined as an enlargement of an object. An enlargement changes the size of an object by multiplying each of the lengths by a scale factor to make it larger or smaller. The scale of a drawing is usually stated as a ratio.

Now, the scale factor of the given drawing is seen as 2 in : 5 ft

From the drawing the dimensions of the kitchen are:

9" x 7"

Thus:

True length = (9 * 5)/2 = 22.5 ft

True width = (7 * 5)/2 = 17.5 ft

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By using the formula for variance

[tex]varX= m*(n*(m-1)/m^n)(1 - n(m-1)/(m^n-1))[/tex]

To compute varX:

we first need to find the expected value of X, denoted as E(X).

We can approach this by using the linearity of expectation, which states that the expected value of the sum of random variables is equal to the sum of their individual expected values.

Let's define a random variable Xi as the number of bins with exactly one ball. Then, we have:

[tex]X = X1 + X2 + ... + Xm[/tex]

where m is the total number of bins.

By the definition of Xi, we know that Xi can only take on values between 0 and 1, since a bin can either have exactly one ball (Xi = 1) or not (Xi = 0).

To find E(Xi), we can use the probability of Xi being 1. The probability that a specific bin has exactly one ball is given by:

[tex]P(Xi = 1) = (n choose 1) * ((m-1) choose (n-1)) / (m choose n)[/tex]

The first term (n choose 1) represents the number of ways to choose one ball out of n balls to put into the bin. The second term ((m-1) choose (n-1)) represents the number of ways to choose (n-1) balls out of the remaining (m-1) bins. Dividing by (m choose n) gives us the probability that exactly one bin has one ball.

Therefore, we have:

E(Xi) = P(Xi = 1) * 1 + P(Xi = 0) * 0
     = P(Xi = 1)=[tex](n choose 1) * ((m-1) choose (n-1)) / (m choose n)[/tex]
Using the linearity of expectation, we can find E(X) as:

E(X) = E(X1) + E(X2) + ... + E(Xm)
    = [tex]m * (n choose 1) * ((m-1) choose (n-1)) / (m choose n)[/tex]

Now, to find varX, we need to find the variance of Xi and use the formula for variance of a sum of random variables.

The variance of Xi can be found as:

Var(Xi) = E(Xi^2) - (E(Xi))^2

Since Xi can only take on values 0 or 1, we have:

E(Xi^2) =[tex]0^2 * P(Xi = 0) + 1^2 * P(Xi = 1) = P(Xi = 1)[/tex]

Therefore, we have:

Var(Xi) = P(Xi = 1) - (E(Xi))^2
      = [tex]m*(n*(m-1)/m^n) + m*(m-1)(n(m-1)/m^n)^2 - (mn(m-1)/m^n)^2[/tex]

Using the formula for variance of a sum of random variables, we have:

varX = Var(X1 + X2 + ... + Xm)
    = Var(X1) + Var(X2) + ... + Var(Xm)      (since Xi's are independent)
    = [tex]m*(n*(m-1)/m^n)(1 - n(m-1)/(m^n-1))[/tex]

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The predicted number of guests at the festival in 2022 is 1846.15, rounded to two decimal places.

To calculate the 2-year Weighted Moving Average (WMA) to predict the number of guests at the festival in 2022, you will need to follow these steps:

1. First, you need to calculate the weighted average for the most recent two years. To do this, you multiply the number of guests in 2021 (2968) by the first weight (0.7), and you multiply the number of guests in 2020 (1657) by the second weight (0.3). Then you add the two products together to get the weighted average for 2020-2021:

Weighted Average = (2968 x 0.7) + (1657 x 0.3) = 2077.9

2. Next, you need to calculate the weighted average for the previous two years. To do this, you multiply the number of guests in 2019 (1559) by the first weight (0.7), and you multiply the number of guests in 2018 (1771) by the second weight (0.3). Then you add the two products together to get the weighted average for 2018-2019:

Weighted Average = (1559 x 0.7) + (1771 x 0.3) = 1614.4

3. Finally, you take the average of the two weighted averages calculated in steps 1 and 2 to get the 2-year Weighted Moving Average for 2022:

2-year WMA for 2022 = (2077.9 + 1614.4) / 2 = 1846.15

Therefore, the predicted number of guests at the festival in 2022 is 1846.15, rounded to two decimal places.

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Answer: Y = 45x + 140

Step-by-step explanation:

which equation represents the length of the completed tunnel based on the number of days since tbm was introduced? the answer is y = 45x + 140

Answer: x=67    x=70    x=61

Step-by-step explanation:

see image for explanaton

The conditions that must hold for inferential procedures to be valid for this scenario are:

(b) the two sample groups must be independent of each other, (c) the data distribution for both men's and women's hemoglobin levels must be normally distributed or each sample size must be larger than 30, and (d) the observations within each sample must be independent.

(b) The two sample groups must be independent of each other because the samples should not be related in any way that could influence the results.

(c) The data distribution for both men's and women's hemoglobin levels must be normally distributed or each sample size must be larger than 30. If the data is not normally distributed, then the Central Limit Theorem can be applied if the sample size is greater than 30.

(d) The observations within each sample must be independent to avoid bias in the results. This means that each observation should not be influenced by any other observation.

(a) The expected count for each level of the categorical variable must be at least 5. This condition is relevant for Chi-square tests of independence, which are not being used in this scenario.

(e) There must be 10 successes and 10 failures in each sample. This condition is relevant for tests of proportions, which are not being used in this scenario.

So b,c and d are correct options.

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The solution to the equation ex + x = 7 using Newton's method with an initial guess of Xo = 2 is approximately 1.4449.

(a) To determine the equation y = a + mx of the least square line that best fits the given data points (2,1), (1,1), (3,2), we first need to calculate the mean of x and y, and then calculate the slope m and y-intercept a of the line.

Mean of x: (2 + 1 + 3)/3 = 2

Mean of y: (1 + 1 + 2)/3 = 4/3

To calculate the slope m, we need to find the sum of (xi - x-bar)(yi - y-bar) and the sum of (xi - x-bar)^2 for each data point:

(2-2)(1-4/3) + (1-2)(1-4/3) + (3-2)(2-4/3) = -1/3

(2-2)^2 + (1-2)^2 + (3-2)^2 = 6

So, m = (-1/3) / 6 = -1/18

To calculate the y-intercept a, we can use the formula a = y-bar - m(x-bar):

a = (4/3) - (-1/18)(2) = 5/9

Therefore, the equation of the least square line is y = (5/9) - (1/18)x.

(b) We want to solve the equation ex + x = 7 using Newton's method with an initial guess of Xo = 2.

First, we need to find the derivative of the function f(x) = ex + x:

f'(x) = ex + 1

Then, we can use the formula for Newton's method:

Xn+1 = Xn - f(Xn) / f'(Xn)

Plugging in X0 = 2 and using four significant digits:

X1 = 2 - (e^2 + 2) / (e^2 + 1) = 1.574

X2 = 1.574 - (e^1.574 + 1.574) / (e^1.574 + 1) = 1.4633

X3 = 1.4633 - (e^1.4633 + 1.4633) / (e^1.4633 + 1) = 1.445

X4 = 1.445 - (e^1.445 + 1.445) / (e^1.445 + 1) = 1.4449

Therefore, the solution to the equation ex + x = 7 using Newton's method with an initial guess of Xo = 2 is approximately 1.4449.

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Roots of a quadratic equation using the factorisation method, we need to find two numbers that multiply to the constant term of the equation and add up to the coefficient of the linear term. Then, we can use these two numbers to factor the quadratic expression and solve for the roots.

a) For the quadratic equation x^2 - 10x + 16 = 0, we need to find two numbers that multiply to 16 and add up to -10. These numbers are -2 and -8, so we can write the quadratic as (x - 2)(x - 8) = 0. Setting each factor equal to zero, we get x - 2 = 0 and x - 8 = 0, which give us the roots x = 2 and x = 8.

b) For the quadratic equation 2x^2 + 3x - 9 = 0, we need to find two numbers that multiply to -18 (since 2*(-9) = -18) and add up to 3. These numbers are 6 and -3, so we can write the quadratic as 2x^2 + 6x - 9x - 9 = 0. Factoring by grouping, we get 2x(x + 3) - 9(x + 3) = 0, which simplifies to (2x - 9)(x + 3) = 0. Setting each factor equal to zero, we get 2x - 9 = 0 and x + 3 = 0, which give us the roots x = 9/2 and x = -3.

c) For the quadratic equation x^2 - 5x = 0, we can factor out an x to get x(x - 5) = 0. Setting each factor equal to zero, we get x = 0 and x - 5 = 0, which give us the roots x = 0 and x = 5.

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The initial mortality rate in 1998 per 100,000 people is :

88.8 deaths

To find the initial mortality rate in 1998, you'll need to use the given relation :

M = 88.8(0.9418)^t, where M is the number of deaths per 100,000 people, and t is the number of years since 1998.

Identify the value of t for 1998. Since 1998 is the starting year, t = 0.

Substitute the value of t into the equation. M = 88.8(0.9418)^0

Calculate M. Since any number raised to the power of 0 is 1, the equation becomes M = 88.8(1), which simplifies to M = 88.8.

So, the initial mortality rate in 1998 is 88.8 deaths per 100,000 people.

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The final amount after 20 years, compounded quarterly is $6,353.98.

The final amount after 5 years, compounded continuously, is  $1,145.10.

we can use the formula for compound interest:

[tex]A = P(1 + r/n)^(^n^\times^t^)[/tex]

where A is the final amount,

P is the principal (starting amount),

r is the annual interest rate, t is the time in years, and n is the number of times compounded per year.

Plugging in the given values, we get:

[tex]A = 3750(1 + 0.035/4)^(^4^\times^2^0^)[/tex]

A = $6,353.98

Therefore, the final amount after 20 years, compounded quarterly, is approximately $6,353.98.

P= $1,000; r=2.8%, t= 5yrs compounded continuously ​

For the second problem, we can use the formula for continuous compounding:

[tex]A = Pe^(^r^t^)[/tex]

[tex]A = 1000e^(^0^.^0^2^8^\times^5^)[/tex]

A = $1,145.10

Therefore, the final amount after 5 years, compounded continuously, is  $1,145.10.

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The function is not differentiable at two points, C=1 and C=2, making the answer (d) two.

The given function involves four absolute value terms. To determine the points where the function is not differentiable, we need to check where the absolute value terms change their behavior.

The term |2c| is differentiable everywhere since it always yields a non-negative value, irrespective of the value of c.

The term |2C-2| changes its behavior at C=1, where it changes from decreasing to increasing. The function is not differentiable at C=1. The term |2x-3| is differentiable everywhere.

The term |2C-4| changes its behavior at C=2, where it changes from decreasing to increasing. The function is not differentiable at C=2.

The function is not differentiable at the points where the absolute value terms change from decreasing to increasing or vice versa, which results in a sharp corner or a cusp in the graph of the function.

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We would need a sample size of at least 17 mice per group to detect a difference in proportions with a power of 0.8 and a significance level of 0.05.

To determine the sample size needed for a study, we can use power analysis. In this case, we want to detect a difference in proportions between two groups with a significance level of 0.05 and a power of 0.8.

We can use the following formula to calculate the sample size per group:

n = (Z_1-α/2 + Z_1-β)² × (p_1(1-p_1) + p_2(1-p_2)) / (p_1 - p_2)²

where:

Z_1-α/2 is the z-score corresponding to the chosen significance level (0.05/2 = 0.025 for a two-tailed test)

Z_1-β is the z-score corresponding to the chosen power (0.8 in this case)

p_1 is the proportion of wild type mice that develop cancer (0.25)

p_2 is the proportion of mutant mice that develop cancer (0.75)

Plugging in the values, we get:

n = (1.96 + 0.84)² × (0.25(1-0.25) + 0.75(1-0.75)) / (0.75 - 0.25)²

n = 16.48

Therefore, we would need a sample size of at least 17 mice per group to detect a difference in proportions with a power of 0.8 and a significance level of 0.05.

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