Question 1 of 40 | Mathematics homework help

Question 1 of 40

Use the exponential growth model, A = A0ekt, to show that the time it takes a population to double (to grow from A0 to 2A0 ) is given by t = ln 2/k.

A. A0 = A0ekt; ln = ekt; ln 2 = ln ekt; ln 2 = kt; ln 2/k = t

B. 2A0 = A0e; 2= ekt; ln = ln ekt; ln 2 = kt; ln 2/k = t

C. 2A0 = A0ekt; 2= ekt; ln 2 = ln ekt; ln 2 = kt; ln 2/k = t

D. 2A0 = A0ekt; 2 = ekt; ln 1 = ln ekt; ln 2 = kt; ln 2/k = toe

Question 2 of 40

Use properties of logarithms to expand the following logarithmic expression as much as possible.

Logb (√xy3 / z3)

A. 1/2 logb x – 6 logb y + 3 logb z

B. 1/2 logb x – 9 logb y – 3 logb z

C. 1/2 logb x + 3 logb y + 6 logb z

D. 1/2 logb x + 3 logb y – 3 logb

Question 3 of 40

Evaluate the following expression without using a calculator.

Log7 √7

A. 1/4

B. 3/5

C. 1/2

D. 2/7

Question 4 of 40

Consider the model for exponential growth or decay given by A = A0ekt. If k __________, the function models the amount, or size, of a growing entity. If k __________, the function models the amount, or size, of a decaying entity.

A. > 0; < 0

B. = 0; ≠ 0

C. ≥ 0; < 0

D. < 0; ≤ 0

Question 5 of 40

You have $10,000 to invest. One bank pays 5% interest compounded quarterly and a second bank pays 4.5% interest compounded monthly. Use the formula for compound interest to write a function for the balance in each bank at any time t.

A. A = 20,000(1 + (0.06/4))4t; A = 10,000(1 + (0.044/14))12t

B. A = 15,000(1 + (0.07/4))4t; A = 10,000(1 + (0.025/12))12t

C. A = 10,000(1 + (0.05/4))4t; A = 10,000(1 + (0.045/12))12t

D. A = 25,000(1 + (0.05/4))4t; A = 10,000(1 + (0.032/14))12t

Question 6 of 40

The half-life of the radioactive element krypton-91 is 10 seconds. If 16 grams of krypton-91 are initially present, how many grams are present after 10 seconds? 20 seconds?

A. 10 grams after 10 seconds; 6 grams after 20 seconds

B. 12 grams after 10 seconds; 7 grams after 20 seconds

C. 4 grams after 10 seconds; 1 gram after 20 seconds

D. 8 grams after 10 seconds; 4 grams after 20 seconds

Question 7 of 40

Write the following equation in its equivalent exponential form.

5 = logb 32

A. b5 = 32

B. y5 = 32

C. Blog5 = 32

D. Logb = 32

Question 8 of 40

Find the domain of following logarithmic function.

f(x) = log (2 – x)

A. (∞, 4)

B. (∞, -12)

C. (-∞, 2)

D. (-∞, -3)

Question 9 of 40

Approximate the following using a calculator; round your answer to three decimal places.

e-0.95

A. .483

B. 1.287

C. .597

D. .387

Question 10 of 40

Evaluate the following expression without using a calculator.

8log8 19

A. 17

B. 38

C. 24

D. 19

The problem is unclear. My first guess would be that this is meant to be 8 log₈19, but none of the answers match that. My second guess would be that it is actually supposed to be 8 log₉ 81, which would be equal to 16, but none of the answers match with that either.

Can you clarify the given expression?

Question 11 of 40

Write the following equation in its equivalent exponential form.

4 = log2 16

A. 2 log4 = 16

B. 22 = 4

C. 44 = 256

D. 24 = 16

Question 12 of 40

An artifact originally had 16 grams of carbon-14 present. The decay model A = 16e -0.000121t describes the amount of carbon-14 present after t years. How many grams of carbon-14 will be present in 5715 years?

A. Approximately 7 grams

B. Approximately 8 grams

C. Approximately 23 grams

D. Approximately 4 grams

Question 13 of 40

Use properties of logarithms to expand the following logarithmic expression as much as possible.

logb (x2 y) / z2

A. 2 logb x + logb y – 2 logb z

B. 4 logb x – logb y – 2 logb z

C. 2 logb x + 2 logb y + 2 logb z

D. logb x – logb y + 2 logb z

Question 14 of 40

Solve the following exponential equation by expressing each side as a power of the same base and then equating exponents.

ex+1 = 1/e

A. {-3}

B. {-2}

C. {4}

D. {12}

Question 15 of 40

Write the following equation in its equivalent logarithmic form.

2-4 = 1/16

A. Log4 1/16 = 64

B. Log2 1/24 = -4

C. Log2 1/16 = -4

D. Log4 1/16 = 54

Question 16 of 40

Solve the following exponential equation. Express the solution set in terms of natural logarithms or common logarithms to a decimal approximation, of two decimal places, for the solution.

32x + 3x – 2 = 0

A. {1}

B. {-2}

C. {5}

D. {0}

Question 17 of 40

The graph of the exponential function f with base b approaches, but does not touch, the __________-axis. This axis, whose equation is __________, is a __________ asymptote.

A. x; y = 0; horizontal

B. x; y = 1; vertical

C. -x; y = 0; horizontal

D. x; y = -1; vertical

Question 18 of 40

The exponential function f with base b is defined by f(x) = __________, b > 0 and b ≠ 1. Using interval notation, the domain of this function is __________ and the range is __________.

A. bx; (∞, -∞); (1, ∞)

B. bx; (-∞, -∞); (2, ∞)

C. bx; (-∞, ∞); (0, ∞)

D. bx; (-∞, -∞); (-1, ∞)

Question 19 of 40

Solve the following logarithmic equation. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, to two decimal places, for the solution.

2 log x = log 25

A. {12}

B. {5}

C. {-3}

D. {25}

Question 20 of 40

Approximate the following using a calculator; round your answer to three decimal places.

3√5

A. .765

B. 14297

C. 11.494

D. 11.665

Question 21 of 40

Solve the following system.

2x + y = 2

x + y – z = 4

3x + 2y + z = 0

A. {(2, 1, 4)}

B. {(1, 0, -3)}

C. {(0, 0, -2)}

D. {(3, 2, -1)}

Question 22 of 40

Solve the following system.

x = y + 4

3x + 7y = -18

A. {(2, -1)}

B. {(1, 4)}

C. {(2, -5)}

D. {(1, -3)}

Question 23 of 40

On your next vacation, you will divide lodging between large resorts and small inns. Let x represent the number of nights spent in large resorts. Let y represent the number of nights spent in small inns.

Write a system of inequalities that models the following conditions:

You want to stay at least 5 nights. At least one night should be spent at a large resort. Large resorts average $200 per night and small inns average $100 per night. Your budget permits no more than $700 for lodging.

y ≥ 1

x + y ≥ 5

x ≥ 1

300x + 200y ≤ 700

y ≥ 0

x + y ≥ 3

x ≥ 0

200x + 200y ≤ 700

y ≥ 1

x + y ≥ 4

x ≥ 2

500x + 100y ≤ 700

y ≥ 0

x + y ≥ 5

x ≥ 1

200x + 100y ≤ 700

Question 24 of 40

Solve the following system by the substitution method.

{x + 3y = 8

{y = 2x – 9

A. {(5, 1)}

B. {(4, 3)}

C. {(7, 2)}

D. {(4, 3)}

Question 25 of 40

Solve the following system.

3(2x+y) + 5z = -1

2(x – 3y + 4z) = -9

4(1 + x) = -3(z – 3y)

A. {(1, 1/3, 0)}

B. {(1/4, 1/3, -2)}

C. {(1/3, 1/5, -1)}

D. {(1/2, 1/3, -1)}

Question 26 of 40

2x + 4y + 3z = 2

x + 2y – z = 0

4x + y – z = 6

A. {(-3, 2, 6)}

B. {(4, 8, -3)}

C. {(3, 1, 5)}

D. {(1, 4, -1)}

None of the given solutions works with this system of equations. Is there possibly a typo somewhere?

Question 27 of 40

Find the quadratic function y = ax2 + bx + c whose graph passes through the given points.

(-1, 6), (1, 4), (2, 9)

A. y = 2×2 – x + 3

B. y = 2×2 + x2 + 9

C. y = 3×2 – x – 4

D. y = 2×2 + 2x + 4

Question 28 of 40

Solve the following system by the substitution method.

{x + y = 4

{y = 3x

A. {(1, 4)}

B. {(3, 3)}

C. {(1, 3)}

D. {(6, 1)}

Question 29 of 40

Find the quadratic function y = ax2 + bx + c whose graph passes through the given points.

(-1, -4), (1, -2), (2, 5)

A. y = 2×2 + x – 6

B. y = 2×2 + 2x – 4

C. y = 2×2 + 2x + 3

D. y = 2×2 + x – 5

Question 30 of 40

Write the partial fraction decomposition for the following rational expression.

x + 4/x2(x + 4)

A. 1/3x + 1/x2 – x + 5/4(x2 + 4)

B. 1/5x + 1/x2 – x + 4/4(x2 + 6)

C. 1/4x + 1/x2 – x + 4/4(x2 + 4)

D. 1/3x + 1/x2 – x + 3/4(x2 + 5)

None of the given solutions matches with the correct solution. The expression (x + 4)/x²(x + 4) reduces to 1/x²

Question 31 of 40

Solve each equation by the substitution method.

x + y = 1

x2 + xy – y2 = -5

A. {(4, -3), (-1, 2)}

B. {(2, -3), (-1, 6)}

C. {(-4, -3), (-1, 3)}

D. {(2, -3), (-1, -2)}

Question 32 of 40

Solve each equation by the substitution method.

x2 – 4y2 = -7

3×2 + y2 = 31

A. {(2, 2), (3, -2), (-1, 2), (-4, -2)}

B. {(7, 2), (3, -2), (-4, 2), (-3, -1)}

C. {(4, 2), (3, -2), (-5, 2), (-2, -2)}

D. {(3, 2), (3, -2), (-3, 2), (-3, -2)}

Question 33 of 40

Write the partial fraction decomposition for the following rational expression.

ax +b/(x – c)2 (c ≠ 0)

A. a/a – c +ac + b/(x – c)2

B. a/b – c +ac + b/(x – c)

C. a/a – b +ac + c/(x – c)2

D. a/a – b +ac + b/(x – c)

Note: Choice A is not actually correct, but it is the closest of the given choices. The partial fraction decomposition for the given expression is:

a/(x-c) + (ac + b)/(x-c)²

Question 34 of 40

A television manufacturer makes rear-projection and plasma televisions. The proﬁt per unit is $125 for the rear-projection televisions and $200 for the plasma televisions.

Let x = the number of rear-projection televisions manufactured in a month and let y = the number of plasma televisions manufactured in a month. Write the objective function that models the total monthly profit.

A. z = 200x + 125y

B. z = 125x + 200y

C. z = 130x + 225y

D. z = -125x + 200y

Question 35 of 40

Many elevators have a capacity of 2000 pounds.

If a child averages 50 pounds and an adult 150 pounds, write an inequality that describes when x children and y adults will cause the elevator to be overloaded.

A. 50x + 150y > 2000

B. 100x + 150y > 1000

C. 70x + 250y > 2000

D. 55x + 150y > 3000

Question 36 of 40

Solve the following system by the addition method.

{4x + 3y = 15

{2x – 5y = 1

A. {(4, 0)}

B. {(2, 1)}

C. {(6, 1)}

D. {(3, 1)}

Question 37 of 40

Solve each equation by either substitution or addition method.

x2 + 4y2 = 20

x + 2y = 6

A. {(5, 2), (-4, 1)}

B. {(4, 2), (3, 1)}

C. {(2, 2), (4, 1)}

D. {(6, 2), (7, 1)}

Question 38 of 40

Write the form of the partial fraction decomposition of the rational expression.

7x – 4/x2 – x – 12

A. 24/7(x – 2) + 26/7(x + 5)

B. 14/7(x – 3) + 20/7(x2 + 3)

C. 24/7(x – 4) + 25/7(x + 3)

D. 22/8(x – 2) + 25/6(x + 4)

Question 39 of 40

Solve the following system.

x + y + z = 6

3x + 4y – 7z = 1

2x – y + 3z = 5

A. {(1, 3, 2)}

B. {(1, 4, 5)}

C. {(1, 2, 1)}

D. {(1, 5, 7)}

Question 40 of 40

Perform the long division and write the partial fraction decomposition of the remainder term.

x5 + 2/x2 – 1

A. x2 + x – 1/2(x + 1) + 4/2(x – 1)

B. x3 + x – 1/2(x + 1) + 3/2(x – 1)

C. x3 + x – 1/6(x – 2) + 3/2(x + 1)

D. x2 + x – 1/2(x + 1) + 4/2(x – 1)

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