Question 1

After t years, the remaining mass y (in grams)of 24 grams of a radioactive elements whose half-life is 30 years is given by $y = 24 \left( \frac { 1 } { 2 } \right) ^ { t / 30 } , t \geq 0$ . How much of the initial mass remains after 50 years? Round your answers to two decimal places.
A)7.56 grams
B)14.40 grams
C)3.17 grams
D)16.44 grams
E)nothing remains

Accepted Answer

To find the remaining mass after 50 years, substitute $t = 50$ into the given formula: $y = 24 \left( \frac { 1 } { 2 } \right) ^ { 50 / 30 }$. This simplifies to $y = 24 \left( \frac { 1 } { 2 } \right) ^ { 5 / 3 }$, which calculates to approximately 7.56 grams.

Question 2

The population P (in millions)of the United States from 1970 to 2000 can be approximated by the exponential function $P ( t ) = 203.0 ( 1.0107 ) ^ { t }$ , where t is the time in years with $t = 0$ corresponding to 1970. Use the model to estimate the population in the year 2,005. Round your answer to one decimal place. Source: U.S. Census Bureau
A)205.2 million
B)7,181.0 million
C)223.9million
D)294.6 million
E)1.5 million

Accepted Answer

To estimate the population in 2005, substitute $t = 35$ (since 2005 is 35 years after 1970) into the function: $P(35) = 203.0 \times (1.0107)^{35}$. This calculation results in approximately 294.6 million.

Question 3

Find $( g \circ f ) ( x )$ where $f ( x ) = 4 x + 5$ and $g ( x ) = x - 2$ .
A) $4 x + 3$
B) $4 x ^ { 2 } - 3 x - 10$
C) $4 x - 3$
D) $5 x + 3$
E) $4 x ^ { 2 } - 8 x + 5$

Accepted Answer

To find $(g \circ f)(x)$, we first apply $f(x)$ and then apply $g(x)$ to the result. So, we start with $f(x) = 4x + 5$. Applying $g(x)$ to this, we get $g(f(x)) = g(4x + 5) = (4x + 5) - 2 = 4x + 3$.

Question 4

Determine the principal P (to the nearest cent)that will yield a balance of A=2,500,000 dollars when invested at rate r=11% for 40 years, compounded once per year. Round the answer to two decimal places.
A)$56,306.31
B)$568,181.82
C)$68,460.62
D)$55,625.00
E)$38,461.03

Accepted Answer

The answer of Determine the principal P (to the nearest...

Question 5

The value of a piece of property doubles every 16 years. You buy the property for $50,000. Its value t years after the date of purchase should be $V ( t ) = 50,000 ( 2 ) ^ { t / 16 }$ . Use the model to approximate (to the nearest thousand dollars)the value of the property after 4 years.
A)$50,000
B)$6,400,000
C)$75,000
D)$59,000
E)$100,000

Accepted Answer

The value of the property after 4 years is $V ( 4 ) = 50,000 ( 2 ) ^ { 4 / 16 } = 50,000 ( 2 ) ^ { 1 / 4 } \approx 59,000$ dollars.

Question 6

Determine the principal P (to the nearest cent)that will yield a balance of $A = 500,000$ dollars when invested at rate $r = 8.5 \%$ for 40 years, compounded continuously. Round the answer to two decimal places.
A)$611,887.72
B)$495,364.63
C)$16,686.63
D)$13,015.58
E)$13,608.96

Accepted Answer

The answer of Determine the principal P (to the nearest...

Question 7

Determine the principal P (to the nearest cent)that will yield a balance of $A = 500,000$ dollars when invested at rate $r = 6.5 \%$ for 40 years, compounded 12 times per year. Round the answer to two decimal places.
A)$973.96
B)$273,927.55
C)$16,025.64
D)$37,398.27
E)$402,834.63

Accepted Answer

The answer of Determine the principal P (to the nearest...

Question 8

Determine the balance A ( to the nearest cent)for P=3,000 dollars invested at a rate r=9% for t=10 , compounded 365 times per year. Round the answer to two decimal places.
A)$9,855.00
B)$8,042.90
C)$7,378.81
D)$6,027.12
E)$7,377.99

Accepted Answer

We can use the formula A = P(1 + r/n)^(nt) where P = 3000, r = 0.09, n = 365, and t = 10. Plugging in these values, we get A = 3000(1 + 0.09/365)^(365*10) ≈ 7377.99. Therefore, the closest answer choice is E.

Question 9

Find $( f \circ g ) ( 4 )$ where $f ( x ) = 5 x + 9$ and $g ( x ) = x - 5$.
A)24
B)-29
C)4
D)28
E)-11

Accepted Answer

The answer of Find \(( f \circ g ) (...

Question 10

Identify the transformation of $f ^ { \prime } ( x ) = 2 ^ { x }$ . $h ( x ) = 2 ^ { x - 6 }$
A)a shift of 6 units downward
B)a shift of 6 units to the left
C)a reflection about the line x=6
D)a shift of 6 units to the right
E)a shift of 6 units upward

Accepted Answer

The original function is a basic exponential function where the base is 2 and the exponent is x. The transformed function is also an exponential function with the same base, but the exponent is now (x-6). This means that the graph of h(x) is the same as the graph of f(x) shifted 6 units to the right. Therefore, the correct answer is D, a shift of 6 units to the right.

Question 11

Find $( g \circ f ) ( 3 )$ where $f ( x ) = 5 x + 2$ and $g ( x ) = x - 5$ .
A)12
B)15
C)-28
D)-8
E)-34

Accepted Answer

First, find $f(3) = 5(3) + 2 = 15 + 2 = 17$. Then, apply $g$ to this result: $g(17) = 17 - 5 = 12$.

Question 12

Find $( f \circ g ) ( x )$ where $f ( x ) = 6 x + 9$ and $g ( x ) = x - 7$ .
A) $6 x + 2$
B) $6 x - 33$
C) $6 x ^ { 2 } - 33 x - 63$
D) $6 x ^ { 2 } - 42 x + 9$
E) $7 x + 33$

Accepted Answer

$(f \circ g)(x) = f(g(x)) = f(x-7) = 6(x-7) + 9 = \boxed{\textbf{(B) }6x-33}$

Question 13

Determine the principal P (to the nearest cent)that will yield a balance of $A = 2,500,000$ dollars when invested at rate $r = 10 \%$ for 40 years, compounded 365 times per year. Round the answer to two decimal places.
A)$38,920.73
B)$1,541,095.89
C)$1,712.33
D)$2,472,756.03
E)$45,814.19

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, $r$ 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. Rearranging the formula to solve for $P$, we get $P = \frac{A}{(1 + \frac{r}{n})^{nt}}$. Substituting $A = 2,500,000$, $r = 0.10$, $n = 365$, and $t = 40$ into the formula, we calculate $P$ to be approximately $45,814.19.

Question 14

Determine the balance A ( to the nearest cent)for P=3,000 dollars invested at a rate r=10% for t=10 , compounded continuously. Round the answer to two decimal places.
A)$8,154.85
B)$10,950.00
C)$8,153.73
D)$8,887.56
E)$6,030.15

Accepted Answer

The correct answer is A because the formula for continuously compounded interest is A = Pe^rt, where P is the principal, r is the interest rate, and t is the time. Plugging in the given values, we get A = 3000e^(0.1*10) = 8154.85.

Question 15

Determine the balance A ( to the nearest cent)for P=3,000 dollars invested at a rate r=10% for t=10 , compounded one time per year. Round the answer to two decimal places.
A)$33,000.00
B)$4,781.23
C)$3,000.00
D)$6,000.00
E)$7,781.23

Accepted Answer

The formula for compound interest is A = P(1 + r/n)^(nt). Here, P = 3000, r = 0.10, n = 1, and t = 10. Plugging in these values, A = 3000(1 + 0.10/1)^(1*10) = 3000(1.10)^10 = 3000 * 2.59374 = 7,781.23.

Question 16

Determine the balance A ( to the nearest cent)for P=500 dollars invested at a rate r=7% for t=10 , compounded 12 times per year. Round the answer to two decimal places.
A)$529.94
B)$4,700.00
C)$1,126.10
D)$1,004.83
E)$6,359.33

Accepted Answer

Use the formula $A=P\left(1+\frac{r}{n}ight)^{nt}$, where $P=500$, $r=0.07$, $n=12$, and $t=10$. Plugging in the values, we get $A=500\left(1+\frac{0.07}{12}ight)^{120}\approx1004.83$. Rounding to the nearest cent gives us $\boxed{	extbf{(D)}\ 1004.83}$.

Question 17

Find $( f \circ g ) ( x )$ where $f ( x ) = 2 x + 6$ and $g ( x ) = x ^ { 2 } - 2$ .
A) $2 x ^ { 3 } 12 x ^ { 2 } - 4 x - 12$
B) $2 x ^ { 2 } + 2$
C) $4 x ^ { 2 } + 24 x + 34$
D) $4 x ^ { 2 } + 4$
E) $2 x ^ { 2 } + 4$

Accepted Answer

The answer of Find \(( f \circ g ) (...

Question 18

Determine the balance A ( to the nearest cent)for P=2,500 dollars invested at a rate r=9.5% for t=10 , compounded 4 times per year. Round the answer to two decimal places.
A)$3,161.40
B)$12,000.00
C)$12,645.59
D)$94,298.50
E)$6,392.88

Accepted Answer

The formula for compound interest is A = P(1 + r/n)^(nt). Here, P = 2500, r = 0.095, n = 4, and t = 10. Plugging in these values, A = 2500(1 + 0.095/4)^(4*10) = 2500(1.02375)^40 ≈ 6392.88.

Question 19

Identify the transformation of $f ( x ) = 15 ^ { x }$ . $h ( x ) = - 15 ^ { x }$
A)a reflection about the y -axis
B)a reflection about the x -axis
C)a reflection about the line y=-x
D)a reflection about the line  y=x
E)a reflection about the line x=15

Accepted Answer

The transformation is a reflection about the x-axis because the negative sign in front of the function causes a reflection of the graph about the x-axis.

Question 20

Determine the principal P (to the nearest cent)that will yield a balance of A=2,500,000 dollars when invested at rate r=9% for 40 years, compounded 4 times per year. Round the answer to two decimal places.
A)$173,611.11
B)$14,218.75
C)$1,026,614.38
D)$147,569.44
E)$71,090.14

Accepted Answer

The answer of Determine the principal P (to the nearest...

Quiz 9: Exponential and Logarithmic Functions

After t years, the remaining mass y (in grams) of 24 grams of a radioactive elements whose half-life is 30 years is given by $y = 24 \left( \frac { 1 } { 2 } \right) ^ { t / 30 } , t \geq 0$ . How much of the initial mass remains after 50 years? Round your answers to two decimal places.

Find $( g \circ f ) ( x )$ where $f ( x ) = 4 x + 5$ and $g ( x ) = x - 2$ .

Determine the principal P (to the nearest cent) that will yield a balance of A=2,500,000 dollars when invested at rate r=11% for 40 years, compounded once per year. Round the answer to two decimal places.

The value of a piece of property doubles every 16 years. You buy the property for $50,000. Its value t years after the date of purchase should be $V ( t ) = 50,000 ( 2 ) ^ { t / 16 }$ . Use the model to approximate (to the nearest thousand dollars) the value of the property after 4 years.

Determine the principal P (to the nearest cent) that will yield a balance of $A = 500,000$ dollars when invested at rate $r = 8.5 \%$ for 40 years, compounded continuously. Round the answer to two decimal places.

Determine the principal P (to the nearest cent) that will yield a balance of $A = 500,000$ dollars when invested at rate $r = 6.5 \%$ for 40 years, compounded 12 times per year. Round the answer to two decimal places.

Determine the balance A ( to the nearest cent) for P=3,000 dollars invested at a rate r=9% for t=10 , compounded 365 times per year. Round the answer to two decimal places.

Find $( f \circ g ) ( 4 )$ where $f ( x ) = 5 x + 9$ and $g ( x ) = x - 5$ .

Identify the transformation of $f ^ { \prime } ( x ) = 2 ^ { x }$ . $h ( x ) = 2 ^ { x - 6 }$

Find $( g \circ f ) ( 3 )$ where $f ( x ) = 5 x + 2$ and $g ( x ) = x - 5$ .

Find $( f \circ g ) ( x )$ where $f ( x ) = 6 x + 9$ and $g ( x ) = x - 7$ .

Determine the principal P (to the nearest cent) that will yield a balance of $A = 2,500,000$ dollars when invested at rate $r = 10 \%$ for 40 years, compounded 365 times per year. Round the answer to two decimal places.

Determine the balance A ( to the nearest cent) for P=3,000 dollars invested at a rate r=10% for t=10 , compounded continuously. Round the answer to two decimal places.

Determine the balance A ( to the nearest cent) for P=3,000 dollars invested at a rate r=10% for t=10 , compounded one time per year. Round the answer to two decimal places.

Determine the balance A ( to the nearest cent) for P=500 dollars invested at a rate r=7% for t=10 , compounded 12 times per year. Round the answer to two decimal places.

Find $( f \circ g ) ( x )$ where $f ( x ) = 2 x + 6$ and $g ( x ) = x ^ { 2 } - 2$ .

Determine the balance A ( to the nearest cent) for P=2,500 dollars invested at a rate r=9.5% for t=10 , compounded 4 times per year. Round the answer to two decimal places.

Identify the transformation of $f ( x ) = 15 ^ { x }$ . $h ( x ) = - 15 ^ { x }$

Determine the principal P (to the nearest cent) that will yield a balance of A=2,500,000 dollars when invested at rate r=9% for 40 years, compounded 4 times per year. Round the answer to two decimal places.

Quiz 9: Exponential and Logarithmic Functions

Find (g∘f) (x) ( g \circ f ) ( x ) (g∘f) (x) where f(x) =4x+5f ( x ) = 4 x + 5f(x) =4x+5 and g(x) =x−2g ( x ) = x - 2g(x) =x−2 .

Determine the principal P (to the nearest cent) that will yield a balance of A=2,500,000 dollars when invested at rate r=11% for 40 years, compounded once per year. Round the answer to two decimal places.

Determine the principal P (to the nearest cent) that will yield a balance of A=500,000A = 500,000A=500,000 dollars when invested at rate r=8.5%r = 8.5 \%r=8.5% for 40 years, compounded continuously. Round the answer to two decimal places.

Determine the principal P (to the nearest cent) that will yield a balance of A=500,000A = 500,000A=500,000 dollars when invested at rate r=6.5%r = 6.5 \%r=6.5% for 40 years, compounded 12 times per year. Round the answer to two decimal places.

Determine the balance A ( to the nearest cent) for P=3,000 dollars invested at a rate r=9% for t=10 , compounded 365 times per year. Round the answer to two decimal places.

Find (f∘g) (4) ( f \circ g ) ( 4 ) (f∘g) (4) where f(x) =5x+9f ( x ) = 5 x + 9f(x) =5x+9 and g(x) =x−5g ( x ) = x - 5g(x) =x−5 .

Identify the transformation of f′(x) =2xf ^ { \prime } ( x ) = 2 ^ { x }f′(x) =2x . h(x) =2x−6h ( x ) = 2 ^ { x - 6 }h(x) =2x−6

Find (g∘f) (3) ( g \circ f ) ( 3 ) (g∘f) (3) where f(x) =5x+2f ( x ) = 5 x + 2f(x) =5x+2 and g(x) =x−5g ( x ) = x - 5g(x) =x−5 .

Find (f∘g) (x) ( f \circ g ) ( x ) (f∘g) (x) where f(x) =6x+9f ( x ) = 6 x + 9f(x) =6x+9 and g(x) =x−7g ( x ) = x - 7g(x) =x−7 .

Determine the principal P (to the nearest cent) that will yield a balance of A=2,500,000A = 2,500,000A=2,500,000 dollars when invested at rate r=10%r = 10 \%r=10% for 40 years, compounded 365 times per year. Round the answer to two decimal places.

Determine the balance A ( to the nearest cent) for P=3,000 dollars invested at a rate r=10% for t=10 , compounded continuously. Round the answer to two decimal places.

Determine the balance A ( to the nearest cent) for P=3,000 dollars invested at a rate r=10% for t=10 , compounded one time per year. Round the answer to two decimal places.

Determine the balance A ( to the nearest cent) for P=500 dollars invested at a rate r=7% for t=10 , compounded 12 times per year. Round the answer to two decimal places.

Find (f∘g) (x) ( f \circ g ) ( x ) (f∘g) (x) where f(x) =2x+6f ( x ) = 2 x + 6f(x) =2x+6 and g(x) =x2−2g ( x ) = x ^ { 2 } - 2g(x) =x2−2 .

Determine the balance A ( to the nearest cent) for P=2,500 dollars invested at a rate r=9.5% for t=10 , compounded 4 times per year. Round the answer to two decimal places.

Identify the transformation of f(x) =15xf ( x ) = 15 ^ { x }f(x) =15x . h(x) =−15xh ( x ) = - 15 ^ { x }h(x) =−15x

Determine the principal P (to the nearest cent) that will yield a balance of A=2,500,000 dollars when invested at rate r=9% for 40 years, compounded 4 times per year. Round the answer to two decimal places.

Find $( g \circ f ) ( x )$ where $f ( x ) = 4 x + 5$ and $g ( x ) = x - 2$ .

Determine the principal P (to the nearest cent) that will yield a balance of $A = 500,000$ dollars when invested at rate $r = 8.5 \%$ for 40 years, compounded continuously. Round the answer to two decimal places.

Determine the principal P (to the nearest cent) that will yield a balance of $A = 500,000$ dollars when invested at rate $r = 6.5 \%$ for 40 years, compounded 12 times per year. Round the answer to two decimal places.

Find $( f \circ g ) ( 4 )$ where $f ( x ) = 5 x + 9$ and $g ( x ) = x - 5$ .

Identify the transformation of $f ^ { \prime } ( x ) = 2 ^ { x }$ . $h ( x ) = 2 ^ { x - 6 }$

Find $( g \circ f ) ( 3 )$ where $f ( x ) = 5 x + 2$ and $g ( x ) = x - 5$ .

Find $( f \circ g ) ( x )$ where $f ( x ) = 6 x + 9$ and $g ( x ) = x - 7$ .

Determine the principal P (to the nearest cent) that will yield a balance of $A = 2,500,000$ dollars when invested at rate $r = 10 \%$ for 40 years, compounded 365 times per year. Round the answer to two decimal places.

Find $( f \circ g ) ( x )$ where $f ( x ) = 2 x + 6$ and $g ( x ) = x ^ { 2 } - 2$ .

Identify the transformation of $f ( x ) = 15 ^ { x }$ . $h ( x ) = - 15 ^ { x }$