Evaluate the function at the specified points.

f(x,y)=x+yx^5, (-1,-3),(-2,4),(2,-2)

At (-1,-3):

At (-2,4):

At (2,-2):

The 74th apple will be placed in second basket of apples.

Apples are distributed, one at a time, into six baskets. the first apple goes into basket one, the second into basket two, the third into basket three, and so on, until each basket has one apple.  

This pattern is repeated, beginning each time with basket one

Since there are 6 baskets, the pattern of distribution repeats every 6 apples.

Hence, to determine which basket the 74th apple will be placed in, we need to find the remainder when 74 is divided by 6:

=> 74 ÷ 6 = 12 remainder 2

This means that the 74th apple will be placed in the second basket, since it follows the pattern of distributing apples starting with the first basket.

Therefore,

The 74th apple will be placed in second basket of apples.

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The integral of sin^4(8q)cos(8q) with respect to q is (1/40)sin^5(8q) + C or (3/8)cos(32q) - (1/64)cos(16q) + C, where C is the constant of integration.

Using the product-to-sum identity for cosine, we can rewrite the integrand as sin(2x+2x+2x+2x)cos(2x+2x) = [sin(2x+2x)cos(2x+2x) + sin(2x)cos(2x+2x+2x)]cos(2x+2x).

We can then use the double angle formula for sine and cosine to simplify the integrand to (3/8)sin(16x) - (1/8)sin(8x) + C. Therefore, the integral of sin4(8q)cos(8q) is (3/8)cos(16q) - (1/64)cos(8q) + C.

To evaluate the integral ∫sin4(8q)cos(8q)dq, we start by using the product-to-sum identity for cosine:

cos(a)sin(b) = 1/2[sin(a+b) + sin(a-b)]

We can rewrite the integrand as:

sin(8q)cos(8q)sin(8q)cos(8q) = [sin(8q+8q)cos(8q+8q) + sin(8q)cos(8q+8q+8q)]cos(8q+8q)

Using the double angle formula for sine and cosine, we can simplify the first term as:

sin(16q)cos(16q) = (1/2)sin(2*16q) = (1/2)sin(32q)

For the second term, we can apply the product-to-sum identity for sine:

sin(a)cos(b) = 1/2[sin(a+b) - sin(a-b)]

sin(8q)cos(8q+8q+8q) = 1/2[sin(8q+24q) - sin(8q-16q)] = 1/2[sin(32q) + sin(8q)]

Putting everything together, we have:

∫sin4(8q)cos(8q)dq = ∫[sin(16q)/2 + sin(32q)/2 + sin(8q)/2]cos(16q)dq

Using the substitution u = 16q, we have:

(1/16)∫[sin(u)/2 + sin(2u)/2 + sin(u/2)/2]cos(u)du

We can then integrate each term separately:

∫sin(u)cos(u)du = (1/2)sin^2(u) + C1

∫sin(2u)cos(u)du = (1/2)[(1/2)sin(3u)] + C2

∫sin(u/2)cos(u)du = -2cos(u/2) + C3

Substituting back, we get:

(1/16)[(1/2)sin^2(16q) + (1/4)sin^2(32q) - 2cos(8q) + C4]

Simplifying, we get:

(3/8)sin^2(16q) - (1/8)sin^2(8q) + C5

Using the identity sin^2(x) = (1-cos(2x))/2, we can rewrite this as:

(3/8)(1-cos(32q))/2 - (1/8)(1-cos(16q))/2 + C6

= (3/8)

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The cooling rate of the turkey is 0.45°F per minute. This means that every minute, the temperature of the turkey decreases by 0.45°F.

The cooling rate of a roasted turkey can be determined by the rate at which it loses heat to its surroundings. In this case, the temperature of the turkey was 191°F when it was taken out of the oven and placed on a table in a room with a temperature of 75°F. After half an hour, the temperature of the turkey had decreased to 155°F.

To calculate the cooling rate, we can use Newton's law of cooling, which states that the rate of heat loss of an object is proportional to the difference in temperature between the object and its surroundings. The equation for Newton's law of cooling is:

dT/dt = -k (T - Ts)

where dT/dt is the rate of change of temperature with respect to time, T is the temperature of the turkey at time t, Ts is the temperature of the surroundings (75°F), and k is a constant that depends on the specific heat of the turkey, its surface area, and other factors.

To solve for k, we can use the data given:

dT/dt = -k (T - Ts)

-36 = -k (155 - 75)

-36 = -k (80)

k = 0.45

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The length of the curve is approximately 0.873 units.

To find the length of the curve, we use the formula:

L = ∫a^b ||r'(t)|| dt

where r'(t) is the derivative of r(t), and ||r'(t)|| is the magnitude of r'(t).

First, let's find the derivative of r(t):

r'(t) = -5sin(5t) i + 5cos(5t) j - (5sin(t)/cos(t)) k

= -5sin(5t) i + 5cos(5t) j - 5tan(t) k

Next, let's find the magnitude of r'(t):

||r'(t)|| = sqrt[(-5sin(5t))^2 + (5cos(5t))^2 + (-5tan(t))^2]

= sqrt[25 + 25tan^2(t)]

Now, we can find the length of the curve:

L = ∫0^(π/4) sqrt[25 + 25tan^2(t)] dt

To solve this integral, we make the substitution u = tan(t), du/dt = sec^2(t), dt = du/sec^2(t), and rewrite the integral as:

L = ∫0^1 sqrt[25 + 25u^2] du/[(1 + u^2)^(1/2)]

Next, we make the substitution v = u/5, dv = du/5, and rewrite the integral as:

L = 5 ∫0^0.2 sqrt[1 + v^2] dv

Using the formula for the integral of the square root of a quadratic, we get:

L = 5/2 [(1/2)(1 + (0.2)^2)^(3/2) - (1/2)(1 + 0^(2))^(3/2)]

= 5/2 [(1.04)^(3/2) - 1]

≈ 0.873

Therefore, the length of the curve is approximately 0.873 units.

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Taryn saved $13.50 by shopping on tax-free weekend, since she did not have to pay any sales tax on her $180 purchase.

to calculate how much taryn saved by shopping on tax-free weekend, we first need to calculate how much she would have paid in sales tax if she had bought her school supplies on a regular day.

if the sales tax is normally 7.5%, then the amount of sales tax taryn would have paid is:

0.075 x $180 = $13.50 the answer is (b) $13.50.

Taryn bought all her school supplies on tax-free weekend and spent $180. If sales tax is normally 7. 5%,

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There are 96 different possible combos.

The principle used to calculate the number of possible combos is called the multiplication principle.

We have,

(a)

The number of possible combos, when you select a popcorn, a candy, and a soda.

= 8 (choices of candy) x 6 (choices of soda) x 2 (choices of popcorn)

= 96

(b)

The principle used to calculate the number of possible combos is called the multiplication principle.

Thus,

There are 96 different possible combos.

The principle used to calculate the number of possible combos is called the multiplication principle.

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Answer:

9749

Step-by-step explanation:

45000·(1+4%)5-45000

9749.38061

rounded to the nearest penny

9749

The minimum order of the Taylor polynomial centred at O for cos x required to approximate cos (-0.85) with an absolute error no greater than 10-4 is 5.

The Taylor series for cos x centred at O is given by:

cos x = 1 - x^2/2! + x^4/4! - x^6/6! + ...

The nth-order Taylor polynomial for cos x centred at O is given by the first n terms of the Taylor series. We want to find the minimum n such that the absolute error between cos (-0.85) and the nth-order Taylor polynomial is no greater than 10-4.

The error term for the nth-order Taylor polynomial is given by:

Rn(x) = cos (c) * xn+1 / (n+1)!

where c is some value between 0 and x.

To find the minimum n, we need to find the value of n such that the error term is no greater than 10-4 for x = -0.85.

Substituting x = -0.85 into the error term and using the fact that |cos (c)| <= 1, we have:

|Rn(-0.85)| <= |(-0.85)^(n+1) / (n+1)!|

We want to find the minimum n such that the right-hand side is no greater than 10-4.

We can use a computer or calculator to find that n = 5 is the smallest integer that satisfies this condition. Therefore, the minimum order of the Taylor polynomial centred at O for cos x required to approximate cos (-0.85) with an absolute error no greater than 10-4 is 5.

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The expression that is equivalent to (2x² + 7x - 15)/(3x² + 16x + 5) * (3x² - 2x - 1)/(2x² - x - 3) is (D) (x - 1)/(x + 1)

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

(2x² + 7x - 15)/(3x² + 16x + 5) * (3x² - 2x - 1)/(2x² - x - 3)

When the expressions are factored, we have:

(2x² + 7x - 15)/(3x² + 16x + 5) * (3x² - 2x - 1)/(2x² - x - 3) = (2x - 3)(x + 5)/(3x + 1)(x + 5) * (3x + 1)(x - 1)/(x + 1)(2x - 3)

Cancelling out the common factors, we have

(2x² + 7x - 15)/(3x² + 16x + 5) * (3x² - 2x - 1)/(2x² - x - 3) = (x - 1)/(x + 1)

This means that the equivalent expression is (D) (x - 1)/(x + 1)

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Answer:

it will take 9 hours to empty the pool.

Step-by-step explanation:

The perimeter of the triangle is 170 inches.

To solve for the perimeter, we first need to find the length of the perpendicular height. We can do this using the sine function:

sin(35°) = 46/x

x = 46/sin(35°) = 65 inches

Now that we know the length of the perpendicular height, we can find the length of the base of the triangle using the cosine function:

cos(35°) = 65/x

x = 65/cos(35°) = 75 inches

The perimeter of the triangle is the sum of the lengths of the three sides, so the perimeter is:

P = 65 + 75 + 30

= 170 inches

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The differential of the function z = x^6 ln(y^4) is dz = 6x^5 ln(y^4) dx + 4x^6 (1/y) dy.

To find the differential of the function z = x^6 ln(y^4), we use the rules of partial differentiation.

Taking the partial derivative of z with respect to x, we get ∂z/∂x = 6x^5 ln(y^4).

Taking the partial derivative of z with respect to y, we get ∂z/∂y = (4x^6/y) ln(y^4).

Then, using the differential notation, we can write dz = (∂z/∂x) dx + (∂z/∂y) dy.

Substituting the values we calculated for ∂z/∂x and ∂z/∂y, we get dz = 6x^5 ln(y^4) dx + 4x^6 (1/y) dy.

This represents the differential of the function z = x^6 ln(y^4).

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Answer:

0.375 gallons

Step-by-step explanation:

The p-value for the given t-statistic falls between 0.005 and 0.01.

o find the p-value for the given t statistic of -2.63 with degrees of freedom of 24, we need to consult the t-distribution table or use statistical software.

Since the alternative hypothesis is μ < 100, we are conducting a one-tailed test in the left tail of the t-distribution. We want to find the area under the t-distribution curve to the left of -2.63.

Using the t-distribution table or software, we can determine that the p-value falls between 0.005 and 0.01. This means that the p-value for the statistic falls between 0.005 and 0.01.

Therefore, the p-value for the given t-statistic falls between 0.005 and 0.01.

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A standard normal distribution table or a calculator that can perform the calculations based on the given z-scores.

To solve these problems, we'll use the properties of the normal distribution. Let's go through each question step by step:

b. What is the distribution of ¯xx¯? ¯xx¯~ N( ? ), (?)

The average of a sample follows a normal distribution with the same mean as the population and a standard deviation equal to the population standard deviation divided by the square root of the sample size. In this case, the population mean is 65, and the population standard deviation is 17. Since we have 50 randomly selected tires, the sample size is 50.

Therefore, the distribution of the sample mean ¯xx¯ is ¯xx¯~N(65, 17/√50).

c. If a randomly selected individual tire is tested, find the probability that the number of miles (in thousands) before it will need replacement is between 67.4 and 69.6?

To find this probability, we need to standardize the values using the z-score formula:

z = (x - μ) / σ

where x is the value we're interested in, μ is the population mean, and σ is the population standard deviation.

For 67.4:

z1 = (67.4 - 65) / 17

For 69.6:

z2 = (69.6 - 65) / 17

We can now use these z-scores to find the probabilities associated with the values using a standard normal distribution table or a calculator. The probability will be the difference between the two probabilities:

P(67.4 ≤ x ≤ 69.6) = P(z1 ≤ Z ≤ z2)

d. For the 50 tires tested, find the probability that the average miles (in thousands) before the need for replacement is between 67.4 and 69.6?

Since we're dealing with the average of the sample, we use the distribution ¯xx¯~N(65, 17/√50) as calculated in part b.

Again, we'll use the z-score formula to standardize the values:

z1 = (67.4 - 65) / (17 / √50)

z2 = (69.6 - 65) / (17 / √50)

Using these z-scores, we can find the probability:

P(67.4 ≤ ¯xx¯ ≤ 69.6) = P(z1 ≤ Z ≤ z2)

Please note that to obtain the precise probabilities, we would need to use a standard normal distribution table or a calculator that can perform the calculations based on the given z-scores.

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Thus, we can find nonzero matrices a, b, and c such that ac = bc and a is not equal to b by using the distributive property of matrix multiplication. One example of such matrices is provided above.

The problem statement requires us to find three matrices - a, b, and c, such that their product ac is equal to bc but a is not equal to b. To solve this problem, we need to use the properties of matrix multiplication. One such property is the distributive property, which states that a(b + c) = ab + ac, where a, b, and c are matrices.

Let's assume that a, b, and c are all 2x2 matrices. One example of such matrices could be:

a = [1 0]
   [0 2]

b = [2 0]
   [0 1]

c = [1 2]
   [3 4]

Using these matrices, we can verify that ac = bc, as follows:

ac = [1 0]  [1 2] = [1 2]
    [0 2]  [3 4]   [6 8]

bc = [2 0]  [1 2] = [2 4]
    [0 1]  [3 4]   [3 4]

As we can see, both products result in the same matrix. However, a and b are not equal, as a(1,1) = 1 and b(1,1) = 2. Therefore, we have found an example of three nonzero matrices such that ac = bc but a is not equal to b.

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The equation of the tangent line:

y = (-x - sqrt(2)/2) - (π/2)(x - sqrt(2)/2)

To find the equation of the line tangent to the curve at the point corresponding to t = π/4, we need to find the first derivatives of x and y with respect to t, evaluate them at t = π/4, and then use these values to find the slope of the tangent line.

The first derivative of x with respect to t is:

dx/dt = -sint + tcost

The first derivative of y with respect to t is:

dy/dt = cost + tsint

Evaluating these at t = π/4, we get:

dx/dt|t=π/4 = -sqrt(2)/2

dy/dt|t=π/4 = (sqrt(2)/2) + (π/4)(sqrt(2)/2)

The slope of the tangent line is the ratio of the change in y to the change in x. So, the slope of the tangent line at t = π/4 is:

m = dy/dt|t=π/4 / dx/dt|t=π/4 = -1 - π/2

Now, we can use the point-slope form of the equation of a line to find the equation of the tangent line. Using the point (x(π/4), y(π/4)) = (sqrt(2)/2, sqrt(2)/2), we get:

y - (sqrt(2)/2) = (m)(x - sqrt(2)/2)

Substituting the value of m, we get:

y - (sqrt(2)/2) = (-1 - π/2)(x - sqrt(2)/2)

Expanding and simplifying, we get the equation of the tangent line:

y = (-x - sqrt(2)/2) - (π/2)(x - sqrt(2)/2)

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The given order of integration is dx first, then dy, and finally dz. To change the order of integration to dz first, then dx, and finally dy, we need to identify the new limits of integration. We can do this by using the given limits of integration and setting up the new integrals. The equivalent integral with the new order of integration is ∫0^1 ∫0^2x ∫0^20 f(x,y,z)dzdxdy.

To change the order of integration, we need to identify the new limits of integration. We can do this by looking at the given limits of integration and setting up the new integrals. First, we need to integrate with respect to z, so we set the limits of integration for z from 0 to 1 - 2x.

Next, we integrate with respect to x, so we set the limits of integration for x from 0 to 2y. Finally, we integrate with respect to y, so we set the limits of integration for y from 0 to 1/4.

Putting it all together, we get the equivalent integral with the new order of integration: ∫0^1 ∫0^2x ∫0^20 f(x,y,z)dzdxdy.

To change the order of integration, we need to identify the new limits of integration. We can do this by setting up the new integrals based on the given limits of integration. The equivalent integral with the new order of integration is ∫0^1 ∫0^2x ∫0^20 f(x,y,z)dzdxdy.

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The value of the car in the year 2017 will be $4973.

Given that a car's value is decreasing, it had a value of $38,000 in the year 2007.

By 2013, the value had depreciated to $11,000.

The car’s value continues to drop by the same percentage, we need to find the price of the car in year 2017.

So, 2013-2007 = 6 years

Using the exponential decay formula,

P = P₀(1-r)ⁿ

11000 = 38000(1-r)⁶

0.29 = (1-r)⁶

Taking log to both sides,

㏒(0.29) = 6 ㏒(1-r)

-0.53 / 6 = ㏒(1-r)

-0.089 = ㏒(1-r)

r = 18%

Now, 2017 - 2013 = 4

So,

P = 11000(1-0.18)⁴

P = 4973.33

Hence the value of the car in the year 2017 will be $4973.

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Answer:

X= 95 Y=55

Step-by-step explanation:

95X5= 475

2X55= 110

475+110= 585

I hope this helps! : )

Therefore, The double integral in polar coordinates involves integrating the function that defines the height of the solid over the region of the polar plane that defines the base of the solid.

To find the volume V of the solid bounded by the graphs of the equations using a double integral in polar coordinates, we first need to determine the limits of integration. This can be done by finding the intersection points of the curves. Once we have the limits, we can set up the integral as follows:
V = ∬R f(r,θ) rdrdθ
where R is the region in the polar plane bounded by the curves, and f(r,θ) is the height of the solid at each point (r,θ). We then evaluate the integral using the appropriate limits to obtain the volume of the solid.
The double integral in polar coordinates allows us to calculate the volume of a three-dimensional solid bounded by two or more surfaces defined in polar coordinates. It involves integrating the function that defines the height of the solid over the region of the polar plane that defines the base of the solid. The limits of integration are determined by finding the intersection points of the curves that bound the region. Once the limits are established, the integral is evaluated to find the volume of the solid.

Therefore, The double integral in polar coordinates involves integrating the function that defines the height of the solid over the region of the polar plane that defines the base of the solid.

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To draw a line and measure it to the nearest quarter inch, you will need a ruler or tape measure marked in inches.

Place the ruler or tape measure at one end of the line and align it so that the 0 mark lines up with the beginning of the line. Then, count the number of quarter inches to the end of the line and record the measurement.

Measuring to the nearest quarter inch means that you are rounding the measurement to the nearest multiple of 0.25 inches. For example, if the line measures between 3 and 3.24 inches, it would be rounded down to 3 inches; if it measures between 3.25 and 3.49 inches, it would be rounded up to 3.5 inches. This level of precision is commonly used in construction, woodworking, and other fields where precise measurements are important.

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To determine the number of passes required to sort the elements using insertion sort, we need to first understand how insertion sort works. It involves comparing each element with the previous elements in the list and inserting it into the correct position.

So for this particular list of elements: 14, 12, 16, 6, 3, 10, we can see that the first pass would involve comparing the second element (12) with the first element (14) and swapping them to get: 12, 14, 16, 6, 3, 10.

The second pass would involve comparing the third element (16) with the second element (14) and leaving it in place, then comparing it with the first element (12) and swapping them to get: 12, 14, 16, 6, 3, 10.

Similarly, the third pass would involve comparing the fourth element (6) with the previous elements and inserting it into the correct position, resulting in: 6, 12, 14, 16, 3, 10.

The fourth pass would involve comparing the fifth element (3) with the previous elements and inserting it into the correct position, resulting in: 3, 6, 12, 14, 16, 10.

Finally, the fifth pass would involve comparing the sixth element (10) with the previous elements and inserting it into the correct position, resulting in the fully sorted list: 3, 6, 10, 12, 14, 16.

Therefore, the answer to this question would be 5, which is one of the answer choices given.

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Answer: To compare the final amounts, we need to calculate the compound interest for Brianna and the simple interest for Audra.

For compound interest, the formula to calculate the future value is:

A = P(1 + r/n)^(nt)

Where:

A = the future value/amount

P = the principal amount (initial investment)

r = the annual interest rate (as a decimal)

n = the number of times that interest is compounded per year

t = the number of years

For Brianna:

P = $16,500

r = 8% = 0.08

n = 1 (compounded annually)

t = 13 years

A = 16500(1 + 0.08/1)^(1*13)

A = 16500(1.08)^13

A ≈ $42,159.84

After 13 years, Brianna will have approximately $42,159.84.

For simple interest, the formula to calculate the future value is:

A = P(1 + rt)

Where:

A = the future value/amount

P = the principal amount (initial investment)

r = the annual interest rate (as a decimal)

t = the number of years

For Audra:

P = $16,500

r = 8% = 0.08

t = 13 years

A = 16500(1 + 0.08*13)

A = 16500(1 + 1.04)

A ≈ $39,720

After 13 years, Audra will have approximately $39,720.

To determine who will have more and by how much, we subtract Audra's amount from Brianna's amount:

Difference = Brianna's amount - Audra's amount

Difference = $42,159.84 - $39,720

Difference ≈ $2,439.84

Therefore, at the end of 13 years, Brianna will have approximately $2,439.84 more than Audra.

Step-by-step explanation: :)

Answer:

Brianna will have approximately $2,439.84 more than Audra.

hope it helps u

Step-by-step explanation:

Utilization=  0.90909 or 90.909%

Average number of jobs in the system=10

Average number of jobs in queue = 9.091

End to end response time = 500

Queuing time = 454.545

Probability of 5 or more jobs in the system=0.59049

What is probability?

Probability means possibility of any incident. It is a branch of mathematics which deals with the occurrence of any random event. The value can be  expressed from zero to one. Probability has been introduced in Mathematics to give a  prediction of how likely events are to happen. The meaning of probability is nothing but the extent to which something is likely to happen.

In a M/M/1 model the expected  inter-arrival time is 50 msec and the expected service time is 45 msec.

So the mean rate of arrival (λ) = 1/ 50 = 0.02

The mean service rate(μ) = 1/ 45= 0.022

a) Utilization = λ/μ = 0.02/ 0.022= 0.90909 or 90.909%

b) Average number of jobs in the system= λ/ (μ-λ)

                                                                     = 0.02/(0.022-0.02)

                                                                       = 10

c) Average number of jobs in queue = λ² / (μ(μ-λ))

                                                           = 0.0004/ 0.000044

                                                            = 9.091

d) End to end response time = Average number of time in the system/ arrival rate

                                               = 10/ 0.02

                                              = 500

e) Queuing time = λ/( μ(μ-λ))

                           = 0.02/ 0.000044

                           = 454.545

f) Probability of 5 or more jobs in the system= P(n≥5)= (λ/μ)⁵

                                                                                          = (0.9)⁵

                                                                                         = 0.59049

Hence,

Utilization=  0.90909 or 90.909%

Average number of jobs in the system=10

Average number of jobs in queue = 9.091

End to end response time = 500

Queuing time = 454.545

Probability of 5 or more jobs in the system=0.59049

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Correct question is "a computer system is modeled as a m/m/1 queue. the expected inter-arrival time is 50 msec and the expected service time is 45 msec. calculate the following measures of system performance:

The probability that 4 or more students score 85 percent or higher on the exam is 0.1209 or approximately 12.09 percent.

To solve this problem, we need to use the binomial distribution formula. We know that the probability of each student scoring 85 percent or higher on the exam is p = 0.2 (since 20 percent is equivalent to 85 percent or higher). We also know that n = 10 students took the exam.

Now we need to find the probability that 4 or more students score 85 percent or higher. We can use the binomial probability formula to calculate this:

P(X ≥ 4) = 1 - P(X < 4)

= 1 - (P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3))

= [tex]1 - [(10 choose 0) * 0.2^0 * (0.8)^10 + (10 choose 1) * 0.2^1 * (0.8)^9 + (10 choose 2) * 0.2^2 * (0.8)^8 + (10 choose 3) * 0.2^3 * (0.8)^7][/tex]
= 1 - (0.1074 + 0.2684 + 0.3020 + 0.2013)

= 0.1209

Therefore, the probability that 4 or more students score 85 percent or higher on the exam is 0.1209 or approximately 12.09 percent.

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Here, the outer sum is over the variable $i$ and it ranges from $1$ to $n$. For each value of $i$, the inner sum is over the variable $j$ and it ranges from $1$ to $3$.

The expression $j^2$ is the summand, which is added for each value of $j$.

The given sigma notation is:

n

___

\    j^2

/___

j=1

Expanding this sigma notation, we have:

= 1^2 + 2^2 + 3^2 + ... + (n-1)^2 + n^2

= (1 + 4 + 9 + ... + (n-1)^2) + n^2

The sum of squares up to n-1 can be expressed using the formula:

1^2 + 2^2 + 3^2 + ... + (n-1)^2 = n(n-1)(2n-1)/6

Substituting this in the above expression, we get:

= n(n-1)(2n-1)/6 + n^2

= (2n^3 - 3n^2 + n)/6

Therefore, the expanded form of the given sigma notation is (2n^3 - 3n^2 + n)/6.

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Answer:

Step-by-step explanation:

Side A has a length of 80 ft, side b has a length of 150 ft, and side c (the hypotenuse) has a length of 170 ft. Side A will represent a in the pythagorean theorem, side B will represent b, and side C (hypotenuse) will represent c in the equation. If the equation holds true, then the triangle is a right triangle.

So, we plug it in. a^2 + b^2 = c^2 becomes (80)^2 + (150)^2 = (170)^2

(80)^2 + (150)^2= 28,900

(170)^2= 28,900

since the answers are the same, we know the equation holds true, and thus the triangle is a right triangle. Hope this helps!!