Question 1

Solve.
-The number of books in a small library increases according to the function B = 7300e0.05t, where t is measured in years. How many books will the library have after 9 years?
A) 2532
B) 5829
C) 20,574
D) 11,449

Accepted Answer

To find the number of books after 9 years, substitute t = 9 into the given function: B = 7300e^(0.05*9). Calculating this gives B ≈ 11,449 books.

Question 2

Solve.
-Use the formula N = Iekt, where N is the number of items at time t, I is the initial amount, and k is a growth constant equal to the percent of growth (expressed in decimal form) per unit of time.
There are currently 70 million cars in a certain country, increasing by 3% annually. How many
Years will it take for this country to have 97 million cars? Round to the nearest year.
A) 110 yr
B) 4 yr
C) 9 yr
D) 11 yr

Accepted Answer

To solve for the number of years it will take for the country to have 97 million cars, we use the formula $N = Ie^{kt}$, where $N = 97$ million, $I = 70$ million, $k = 0.03$ (since 3% growth rate is 0.03 in decimal form), and $t$ is the number of years. Rearranging the formula to solve for $t$, we get $t = \frac{\ln(N/I)}{k}$. Plugging in the values, $t = \frac{\ln(97/70)}{0.03}$, which approximately equals 11 years when rounded to the nearest year.

Question 3

Evaluate the given function.
-$$f ( x ) = 2 ^ { - x } , f ( 9 )$$
A) $- 512$
B) $\frac { 1 } { 512 }$
C) $\frac { 1 } { 18 }$
D) $\frac { 2 } { 9 }$

Accepted Answer

Substitute $9$ for $x$ in the given function to get $f(9)=2^{-9}=\frac{1}{2^9}=\frac{1}{512}$. Therefore, the correct choice is $\boxed{	extbf{(B)}\ \frac{1}{512}}$.

Question 4

Solve.
-Susan purchased a painting in the year 2000 for $3000. Assuming an exponential rate of inflation of 3.8% per year, how much will the painting be worth 7 years later?
A) $7812.90
B) $9176.03
C) $4891.02
D) $3914.21

Accepted Answer

The future value of the painting can be calculated using the formula for exponential growth: Future Value = Present Value * (1 + rate)^time. Here, Present Value = $3000, rate = 0.038, and time = 7 years. Plugging in these values gives: $3000 * (1 + 0.038)^7 = $3914.21.

Question 5

Evaluate the given function.
-$$f ( x ) = 4 ^ { x } , f ( - 3 )$$
A) $\frac { 1 } { 12 }$
B) $\frac { 1 } { 64 }$
C) $- \frac { 4 } { 3 }$
D) 12

Accepted Answer

Plug in -3 for x in the function: 
$f(-3) = 4^{-3} = \frac{1}{4^3} = \frac{1}{64}$

Question 6

Evaluate the given function.
-$$f ( x ) = 3 ^ { x + 1 } + 3 , f ( 3 )$$
A) 84
B) 3
C) 6
D) 30

Accepted Answer

To evaluate $f(3)$, substitute $x = 3$ into the function: $f(3) = 3^{3+1} + 3 = 3^4 + 3 = 81 + 3 = 84$.

Question 7

Compute the compound interest.
-$5000 is invested at 9% compounded quarterly. In how many years will the account have grown to $10,500? Round to the nearest tenth of a year.
A) 8.3 yr
B) 8.6 yr
C) 1.1 yr
D) 12.1 yr

Accepted Answer

The formula for compound interest is $A = P(1 + \frac{r}{n})^{nt}$, 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), $n$ is the number of times that interest is compounded per year, and $t$ is the time the money is invested for in years. Here, $A = 10500$, $P = 5000$, $r = 0.09$ (9% as a decimal), and $n = 4$ (since the interest is compounded quarterly). We need to solve for $t$.Rearranging the formula to solve for $t$, we get:$$t = \frac{\log(A/P)}{n \cdot \log(1 + r/n)}$$Substituting the given values:$$t = \frac{\log(10500/5000)}{4 \cdot \log(1 + 0.09/4)}$$$$t \approx 8.3$$ years.Therefore, it will take approximately 8.3 years for the account to grow to $10,500.

Question 8

Compute the compound interest.
-How long must $4200 be in a bank at 6% compounded annually to become $7521.56? (Round to the nearest year.)
A) 10 yr
B) 11 yr
C) 12 yr
D) 9 yr

Accepted Answer

The answer of Compute the compound interest.
-How long must $4200...

Question 9

Evaluate the given function.
-$$f ( x ) = 2 ^ { x } , f ( 4 )$$
A) 16
B) $\frac { 1 } { 2 }$
C) 8
D) $\frac { 1 } { 8 }$

Accepted Answer

To evaluate $f(4)$ for the function $f(x) = 2^x$, substitute $x = 4$ into the function: $f(4) = 2^4 = 16$.

Question 10

Evaluate the given function.
-$f ( x ) = \left( \frac { 1 } { 3 } \right) ^ { x + 2 } + 7 , f ( 1 )$
A) $7 \frac { 1 } { 2187 }$
B) $7 \frac { 1 } { 9 }$
C) $7 \frac { 1 } { 27 }$
D) $10 \frac { 1 } { 27 }$

Accepted Answer

The answer of Evaluate the given function.
-\(f ( x )...

Question 11

Evaluate the given function.
-$f ( x ) = \left( \frac { 1 } { 5 } \right) ^ { x } , f ( - 3 )$
A) $\frac { 1 } { 125 }$
B) $- 15$
C) $- \frac { 1 } { 125 }$
D) 125

Accepted Answer

The answer of Evaluate the given function.
-\(f ( x )...

Question 12

Solve.
-The function $$\mathrm { A } = \mathrm { A } \cap \mathrm { e } ^ { - 0.0077 \mathrm { x } }$$
A) 900 lb
B) 168 lb
C) 178 lb
D) 588 lb

Accepted Answer

The answer of Solve.
-The function \[\mathrm { A } =...

Question 13

Solve.
-The function A = A0e-0.01155x models the amount in pounds of a particular radioactive material stored in a concrete vault, where x is the number of years since the material was put into the vault.
If 300 pounds of the material are placed in the vault, how much time will need to pass for only 53
Pounds to remain?
A) 160 yr
B) 150 yr
C) 155 yr
D) 300 yr

Accepted Answer

The answer of Solve.
-The function A = A0e-0.01155x models the...

Question 14

Evaluate the given function.
-$f ( x ) = \left( \frac { 1 } { 5 } \right) ^ { x }$
$f ( 2 )$
A) 25
B) $\frac { 1 } { 10 }$
C) $\frac { 1 } { 32 }$
D) $\frac { 1 } { 25 }$

Accepted Answer

Plugging in 2 for x, we get $f(2)=\left(\frac{1}{5}\right)^2=\frac{1}{25}$. Therefore, the answer is D.

Question 15

Solve.
-One method to determine the time since an animal died is to estimate the percentage of carbon-14 remaining in its bones. The percent P in decimal form of carbon-14 remaining x years is given by
P(x) = e-0.000121x. Approximate (to the nearest whole year) the age of a fossil if there is 65% of
Carbon-14 remaining.
A) 1546
B) -34,499
C) -15,469
D) 3560

Accepted Answer

We need to solve for x in the equation:

0.65 = e^(-0.000121x)

Taking the natural logarithm of both sides, we get:

ln(0.65) = -0.000121x

Solving for x, we get:

x = ln(0.65)/(-0.000121) ≈ 3560

So the fossil is approximately 3560 years old, which corresponds to choice D.

Question 16

Compute the compound interest.
-How long will it take for $5000 to grow to $15,500 at an interest rate of 3.1% if the interest is compounded quarterly? Round the number of years to the nearest hundredth.
A) 37.06 yr
B) 100.39 yr
C) 36.64 yr
D) 146.55 yr

Accepted Answer

We can use the formula for compound interest:

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

where:

A = final amount = $15,500
P = starting amount = $5000
r = interest rate = 3.1% = 0.031
n = number of times interest is compounded per year = 4 (quarterly)
t = time in years

Plugging in the values and solving for t, we get:

$15,500 = $5000(1 + 0.031/4)^(4t)
3.1 = (1.00775)^(4t)
ln 3.1 = ln(1.00775)^(4t)
ln 3.1 = 4t ln(1.00775)
t = ln 3.1 / (4 ln(1.00775))
t ≈ 36.64 years

Rounding to the nearest hundredth, we get 36.64 years. Therefore, the answer is C.

Question 17

Compute the compound interest.
-How long will it take for $8900 to grow to $12,400 at an interest rate of 3.5% if the interest is compounded continuously? Round the number of years to the nearest hundredth.
A) 3.5 yr
B) 947.56 yr
C) 0.95 yr
D) 9.48 yr

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 for in years. Rearranging the formula to solve for $t$, we get $t = \frac{\ln(A/P)}{r}$. Plugging in the given values: $A = 12400$, $P = 8900$, $r = 0.035$, we get $t = \frac{\ln(12400/8900)}{0.035}$, which approximately equals 9.48 years.

Question 18

Evaluate the given function.
-$$f ( x ) = 4 ^ { x } + 2 , \quad f ( 2 )$$
A) 256
B) 18
C) 16
D) 10

Accepted Answer

To evaluate $f(2)$, substitute $x = 2$ into the function: $f(2) = 4^2 + 2 = 16 + 2 = 18$.

Question 19

Solve.
-There are currently 51 million cars in a certain country, increasing exponentially by 4.4% annually. How many years will it take for this country to have 60 million cars? Round to the nearest year.
A) 3 yr
B) 4 yr
C) 2 yr
D) 50 yr

Accepted Answer

Let's use the formula for exponential growth:

A = P(1 + r)^t

Where: 
A = final amount (60 million cars) 
P = initial amount (51 million cars) 
r = annual growth rate (4.4% or 0.044 as a decimal) 
t = time (in years)

Plugging in the values, we get:

60,000,000 = 51,000,000(1 + 0.044)^t

Dividing both sides by 51,000,000:

1.1765 = 1.044^t

Using logarithms to solve for t:

log 1.1765 = log 1.044^t

t = log 1.1765 / log 1.044 ≈ 3.96

Rounding to the nearest year, we get 4 years. So it will take about 4 years for the country to have 60 million cars.

Question 20

Solve.
-The population of a small country increases according to the function B = 1,600,000e0.04t, where t is measured in years. How many people will the country have after 2 years?
A) 1,923,623
B) 1,755,056
C) 4,041,166
D) 1,733,259

Accepted Answer

The answer of Solve.
-The population of a small country increases...

Quiz 12: Logarithmic and Exponential Functions

Solve. -The number of books in a small library increases according to the function B = 7300e0.05t, where t is measured in years. How many books will the library have after 9 years?

Evaluate the given function. - $f ( x ) = 2 ^ { - x } , f ( 9 )$

Solve. -Susan purchased a painting in the year 2000 for $3000. Assuming an exponential rate of inflation of 3.8% per year, how much will the painting be worth 7 years later?

Evaluate the given function. - $f ( x ) = 4 ^ { x } , f ( - 3 )$

Evaluate the given function. - $f ( x ) = 3 ^ { x + 1 } + 3 , f ( 3 )$

Compute the compound interest. -$5000 is invested at 9% compounded quarterly. In how many years will the account have grown to $10,500? Round to the nearest tenth of a year.

Compute the compound interest. -How long must $4200 be in a bank at 6% compounded annually to become $7521.56? (Round to the nearest year.)

Evaluate the given function. - $f ( x ) = 2 ^ { x } , f ( 4 )$

Evaluate the given function. - $f ( x ) = \left( \frac { 1 } { 3 } \right) ^ { x + 2 } + 7 , f ( 1 )$

Evaluate the given function. - $f ( x ) = \left( \frac { 1 } { 5 } \right) ^ { x } , f ( - 3 )$

Solve. -The function $\mathrm { A } = \mathrm { A } \cap \mathrm { e } ^ { - 0.0077 \mathrm { x } }$

Evaluate the given function. - $f ( x ) = \left( \frac { 1 } { 5 } \right) ^ { x }$ $f ( 2 )$

Compute the compound interest. -How long will it take for $5000 to grow to $15,500 at an interest rate of 3.1% if the interest is compounded quarterly? Round the number of years to the nearest hundredth.

Compute the compound interest. -How long will it take for $8900 to grow to $12,400 at an interest rate of 3.5% if the interest is compounded continuously? Round the number of years to the nearest hundredth.

Evaluate the given function. - $f ( x ) = 4 ^ { x } + 2 , \quad f ( 2 )$

Solve. -There are currently 51 million cars in a certain country, increasing exponentially by 4.4% annually. How many years will it take for this country to have 60 million cars? Round to the nearest year.

Solve. -The population of a small country increases according to the function B = 1,600,000e0.04t, where t is measured in years. How many people will the country have after 2 years?

Quiz 12: Logarithmic and Exponential Functions

Solve. -The number of books in a small library increases according to the function B = 7300e0.05t, where t is measured in years. How many books will the library have after 9 years?

Evaluate the given function. - f(x) =2−x,f(9) f ( x ) = 2 ^ { - x } , f ( 9 ) f(x) =2−x,f(9)

Solve. -Susan purchased a painting in the year 2000 for $3000. Assuming an exponential rate of inflation of 3.8% per year, how much will the painting be worth 7 years later?

Evaluate the given function. - f(x) =4x,f(−3) f ( x ) = 4 ^ { x } , f ( - 3 ) f(x) =4x,f(−3)

Evaluate the given function. - f(x) =3x+1+3,f(3) f ( x ) = 3 ^ { x + 1 } + 3 , f ( 3 ) f(x) =3x+1+3,f(3)