Question 1

Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
-Explain why  is an exponential function but  is not.

Accepted Answer

The answer of Write the word or phrase that best...

Question 2

Solve.
-At what interest rate must $4800 be compounded annually to equal $7091.79 after 8 yr? (Round to the nearest percent.)

Accepted Answer

The formula for compound interest is A = P(1 + r)^n, where A is the amount of money accumulated after n years, including interest, P is the principal amount, r is the annual interest rate (decimal), and n is the time the money is invested for in years. Rearranging the formula to solve for r gives r = ((A/P)^(1/n)) - 1. Substituting the given values: r = ((7091.79/4800)^(1/8)) - 1 ≈ 0.05, which is 5% when converted to a percentage.

Question 3

Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
-Explain why

Accepted Answer

The answer of Write the word or phrase that best...

Question 4

Solve.
-How long will it take for $8600 to grow to $35,700 at an interest rate of 7.1% if the interest is compounded continuously? Round the number of years to the nearest hundredth.

Accepted Answer

The formula for continuous compounding is $A = Pe^{rt}$, where $A$ is the amount of money accumulated after n years, including interest, $P$ is the principal amount (the initial amount of money), $r$ is the annual interest rate (decimal), and $t$ is the time the money is invested or borrowed for, in years. To find $t$, we rearrange the formula to $t = \frac{\ln(A/P)}{r}$. Substituting the given values, $A = 35700$, $P = 8600$, and $r = 0.071$, we get $t = \frac{\ln(35700/8600)}{0.071}$, which calculates to approximately 20.05 years.

Question 5

Solve.
-The decay of 331 mg of an isotope is given by  where t is time in years. Find the amount left after 9 years.

Accepted Answer

The answer of Solve.
-The decay of 331 mg of an...

Question 6

Solve.
-In the formula A = Iekt, A is the amount of radioactive material remaining from an initial amount I at a given time t and k is a negative constant determined by the nature of the material. A certain radioactive isotope
Decays at a rate of 0.3% annually. Determine the half-life of this isotope, to the nearest year.

Accepted Answer

The half-life of a radioactive isotope is found by setting A = 1/2I in the formula A = Ie^(kt) and solving for t. Given the decay rate of 0.3% annually, k = -0.003. To find the half-life, we set up the equation 1/2 = e^(-0.003t). Taking the natural logarithm of both sides gives ln(1/2) = -0.003t. Solving for t gives t ≈ 231 years.

Question 7

Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
-Explain the error in the following:

Accepted Answer

The answer of Write the word or phrase that best...

Question 8

Solve.
-Suppose the consumption of electricity grows at 4.2% per year, compounded continuously. Find the number of years before the use of electricity has tripled. Round the answer to the nearest hundredth.

Accepted Answer

To find the number of years before the use of electricity has tripled when growing at a rate of 4.2% per year compounded continuously, we use the formula for continuous compounding, $A = Pe^{rt}$, where $A$ is the amount after time $t$, $P$ is the initial amount, $r$ is the rate, and $t$ is the time in years. Since the electricity use is to triple, $A/P = 3$. Substituting the values, we get $3 = e^{0.042t}$. Taking the natural logarithm of both sides gives $ln(3) = 0.042t$. Solving for $t$ gives $t = \frac{ln(3)}{0.042}$, which approximately equals 26.16 years.

Question 9

Solve.
-How long must $5400 be in a bank at 5% compounded annually to become $9697.62? (Round to the nearest year.)

Accepted Answer

The answer of Solve.
-How long must $5400 be in a...

Question 10

Solve.
-The number of acres in a landfill is given by the function  where t is measured in years. How many acres will the landfill have after 4 years? (Round to the nearest acre.)

Accepted Answer

The answer of Solve.
-The number of acres in a landfill...

Question 11

Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
-Explain the error in the following:

Accepted Answer

The answer of Write the word or phrase that best...

Question 12

Solve.
-How long will it take for $7600 to grow to $23,400 at an interest rate of 2.1% if the interest is compounded quarterly? Round the number of years to the nearest hundredth.

Accepted Answer

To find the time, use the compound interest formula: $ A = P(1 + \frac{r}{n})^{nt} $. Here, $ A = 23400 $, $ P = 7600 $, $ r = 0.021 $, and $ n = 4 $. Solving for $ t $, we get $ t \approx 53.69 $ years.

Question 13

Solve.
-Assume the cost of a car is $17,000. With continuous compounding in effect,find the number of years it would take to double the cost of the car at an annual inflation rate of 7.9%. Round the answer to the nearest hundredth.

Accepted Answer

To find the number of years it takes for an investment to double with continuous compounding, you can use the formula $t = \frac{\ln(2)}{r}$, where $t$ is the time in years and $r$ is the annual interest rate (inflation rate in this case) expressed as a decimal. For an inflation rate of 7.9%, $r = 0.079$. Plugging the values into the formula gives $t = \frac{\ln(2)}{0.079} \approx 8.77$ years.

Question 14

Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
-Without using a calculator, explain why  must be greater than 8 but less than 16.

Accepted Answer

The answer of Write the word or phrase that best...

Question 15

Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
-Explain why 1 is excluded from being a logarithmic base.

Accepted Answer

The answer of Write the word or phrase that best...

Question 16

Solve.
-The amount of particulate matter left in solution during a filtering process is given by the equation  , where n is the number of filtering steps. Find the amounts left for n = 0 and n = 5. (Round to the nearest whole number.)

Accepted Answer

The answer of Solve.
-The amount of particulate matter left in...

Question 17

Solve.
-In the formula A = Iekt, A is the amount of radioactive material remaining from an initial amount I at a given time t and k is a negative constant determined by the nature of the material. An artifact is discovered at a certain
Site. If it has 55% of the carbon-14 it originally contained, what is the approximate age of the artifact?
(carbon-14 decays at the rate of 0.0125% annually.) (Round to the nearest year.)

Accepted Answer

Given the decay rate of carbon-14 is 0.0125% annually, we can convert this to a decimal to use in the formula, which gives us k = -0.000125. The formula A = Ie^(kt) can be rearranged to solve for t (time), given that A (the remaining amount of carbon-14) is 55% of the initial amount I. Plugging in the values, we solve for t and round to the nearest year, which gives us approximately 4783 years.

Question 18

Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
-Explain the error in the following:

Accepted Answer

The answer of Write the word or phrase that best...

Question 19

Solve.
-The sales of a mature product (one which has passed its peak) are given by the function  , where t is time in years. Find the sales after 9 years if a  .

Accepted Answer

The answer of Solve.
-The sales of a mature product (one...

Question 20

Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
-Explain how the graph of  could be used to graph the function g(x) = 1 + ln x.

Accepted Answer

The answer of Write the word or phrase that best...

Quiz 13: Exponential Functions and Logarithmic Functions

Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. -Explain why is an exponential function but is not.

Solve. -At what interest rate must $4800 be compounded annually to equal $7091.79 after 8 yr? (Round to the nearest percent.)

Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. -Explain why

Solve. -How long will it take for $8600 to grow to $35,700 at an interest rate of 7.1% if the interest is compounded continuously? Round the number of years to the nearest hundredth.

Solve. -The decay of 331 mg of an isotope is given by where t is time in years. Find the amount left after 9 years.

Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. -Explain the error in the following:

Solve. -Suppose the consumption of electricity grows at 4.2% per year, compounded continuously. Find the number of years before the use of electricity has tripled. Round the answer to the nearest hundredth.

Solve. -How long must $5400 be in a bank at 5% compounded annually to become $9697.62? (Round to the nearest year.)

Solve. -The number of acres in a landfill is given by the function where t is measured in years. How many acres will the landfill have after 4 years? (Round to the nearest acre.)

Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. -Explain the error in the following:

Solve. -How long will it take for $7600 to grow to $23,400 at an interest rate of 2.1% if the interest is compounded quarterly? Round the number of years to the nearest hundredth.

Solve. -Assume the cost of a car is $17,000. With continuous compounding in effect,find the number of years it would take to double the cost of the car at an annual inflation rate of 7.9%. Round the answer to the nearest hundredth.

Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. -Without using a calculator, explain why must be greater than 8 but less than 16.

Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. -Explain why 1 is excluded from being a logarithmic base.

Solve. -The amount of particulate matter left in solution during a filtering process is given by the equation , where n is the number of filtering steps. Find the amounts left for n = 0 and n = 5. (Round to the nearest whole number.)

Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. -Explain the error in the following:

Solve. -The sales of a mature product (one which has passed its peak) are given by the function , where t is time in years. Find the sales after 9 years if a .

Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. -Explain how the graph of could be used to graph the function g(x) = 1 + ln x.