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# 6.8: Chapter Test

Difficulty Level: At Grade Created by: CK-12

Part A

Multiple Choice – Please circle the letter of the correct answer and write that letter in the space provided to the left of each question.

1. ________ What is the $y$-intercept of the graph of $f(x)=4(2)^{x+2}$?
1. 12
2. 48
3. 16
4. 32
2. ________ A bacteria decays such that only one-half of the original amount is present after 48 days. After 192 days, how much of the original 100 g of bacteria will remain?
1. 25 g
2. 6.25 g
3. 3.175 g
4. 12.5 g
3. ________ What is the equation of the horizontal asymptote of the function $y=4(2)^x-3$?
1. $y=3$
2. $y=0$
3. $y=1$
4. $y=-3$
4. ________ What is the solution for $9.6 \times 10^{-5}-5.4 \times 10^{-6}$?
1. $9.06 \times 10^{-5}$
2. $4.2 \times 10^{-5}$
3. $9.06 \times 10^{-6}$
4. $9.06 \times 10^{-6}$
5. ________ If $f(x)=3^x-2x$, what is the value of $f(-1)$?
1. $f(-1)=-\frac{5}{3}$
2. $f(-1)=\frac{7}{3}$
3. $f(-1)=0$
4. $f(-1)=-1$
6. ________ Which one of the following is NOT a growth curve?
1. $y=(4)^x$
2. $y=(3)^{-x}$
3. $y=\left(\frac{7}{6}\right)^x$
4. $y=\left(\frac{3}{4}\right)^x$
7. ________ What is the solution for the exponential equation $8^{3x-4}+7=71$?
1. $x=0$
2. $x=25$
3. $x=2$
4. $x=20$
8. ________ What is the simplified expression for $x^{-2}+y^{-1}$?
1. $\frac{y+x^2}{x^2y}$
2. $\frac{1}{x^2+y}$
3. $(x+y)^{-3}$
4. $y+x^2$
9. ________ If you deposit $1250 into a bank account with an interest rate of 0.75% compounded annually, how much interest would be earned over a period of 15 years? 1.$1398.25
2. $16.70 3.$2448.60
4. \$148.25
10. ________ What is the solution for $(6.8 \times 10^4)\times(7.3 \times 10^5)$?
1. $496.4 \times 10^{10}$
2. $4.964 \times 10^{10}$
3. $49.64 \times 10^{10}$
4. $4.964 \times 10^{9}$
11. ________ What is the value of $\left(\frac{1}{2}\right)^{-3}+9^\frac{1}{2}-64^\frac{2}{3}$?
1. -27
2. 21
3. -5
4. 73
12. ________ Which equation represents the exponential function graphed below?
1. $y=2(3)^x+4$
2. $y=3(2)^x+4$
3. $y=7(2)^x$
4. $y=4(2)^x+3$

Part B

Answer the following questions in the space provided. Show all work.

1. Solve the following exponential equations for ‘$x$'.
1. $\sqrt[3]{\frac{9^{x+1}}{27^x}}=81$
2. $(x-2)^\frac{1}{2}=9^\frac{1}{4}$
2. Simplify each of the following:
1. $\frac{\left(\frac{1}{3}\right)^{-2}+8^\frac{2}{3}+5^{0}}{\left(\frac{1}{49}\right)^\frac{1}{2}}$
2. $\frac{(16^{m-2})(8^{2m+1})}{(4^{x-1})^{-3}}$
3. When you take an aspirin, it slowly dilutes and becomes absorbed by the body at a rate of 30% per hour. If you take a 10 mg capsule at noon, how much of the capsule still remains in your body at 4:00 pm?
4. The new Maytag dishwasher has a digital thermometer that records the internal temperature of the machine during and following its wash cycle so as to prevent burns. The temperature display is updated and recorded every 8 minutes. On the day the company tested the dishwasher, the temperature in the kitchen was $66^\circ F$.

The following table shows the readings that were recorded during the testing of the dishwasher.

Time(min) 0 8 16 24 32 10
Temp$(^\circ F)$ 202 188.4 176.2 165.2 155.2 146.4

(a) Determine the rate at which the dishwasher is cooling.

(b) Determine the value of ‘$a$’ in $y=a(b)^\frac{x}{c}+d$ and write an exponential function to model the temperature of the dishwasher after ‘$t$’ minutes.

(c) Determine the equation of the horizontal asymptote.

(d) Determine the temperature of the dishwasher after 3 hours.

(e) Draw a sketch to represent the dishwasher’s change in temperature over time.

Part 1

1. C
2. B
3. D
4. A
5. B
6. D
7. C
8. A
9. D
10. B
11. C
12. B

Part B

(a)

$& \sqrt[3]{\frac{9^{x+1}}{27^x}}=81 \\& \sqrt[3]{\frac{(3^2)^{x+1}}{(3^3)^x}}=(3)^4 \\& \sqrt[3]{\frac{3^{2x+2}}{3^{3x}}}=(3)^4 \\& \sqrt[3]{3^{-x+2}}=(3)^4 \\& (3^{-x+2})^\frac{1}{3}=(3)^4 \\& (3)^\frac{-1x+2}{3}=(3)^4 \\ & \frac{-1x+2}{3}=4 \\& \cancel{3}\left(\frac{-1x+2}{\cancel{3}}\right)=3(4) \\& -1x+2 =12 \\& -1x+2-2 =12-2 \\ & -1x =10 \\& \frac{\cancel{-1}x}{\cancel{-1}}=\frac{10}{-1} \\& \boxed{x =-10}$

(b)

$&(x-2)^\frac{1}{2}=9^\frac{1}{4} \\& (x-2)^\frac{1}{2}=(3^2)^\frac{1}{4} \\& (x-2)^\frac{1}{2}=(3^{\cancel{2}})^\frac{1}{\underset{2}{\cancel{4}}} \\& (x-2)^\frac{1}{2}=3^\frac{1}{2} \\& x-2=3 \\ & x-2+2=3+2 \\& \boxed{x=5}$

(a)

$&\frac{\left(\frac{1}{3}\right)^{-2}+8^\frac{2}{3}+5^0}{\left(\frac{1}{49}\right)^\frac{1}{2}} \\& \frac{(3)^2+(2^3)^\frac{2}{3}+1}{\sqrt \frac{1}{49}} \\& \frac{9+2^2+1}{\frac{1}{7}} \\ & (9+4+1) \div \frac{1}{7} \\ & \boxed{14 \times \frac{7}{1}=98}$

(b)

$& \frac{(16^{m-2})(8^{2m+1})}{(4^{m-1})^{-3}} \\ & \frac{\left((2^4)^{m-2}\right)\left((2^3)^{2m+1}\right)}{\left((2^2)^{m-1}\right)^{-3}} \\ & \frac{(2^{4m-8})(2^{6m+3})}{(2^{2m-2})^{-3}} \\ & \frac{(2^{4m-8})(2^{6m+3})}{(2^{2m-2})^{-3}} \\ & \frac{(2^{4m-8})(2^{6m+3})}{(2^{-6m+6})} \\& \boxed{2^{4m-8+6m+3+6m-6}=2^{16m-11}}$

1. $y &=a(b)^x \\y &=10(0.70)^4 \\y &=2.4 \ mg$

After 4 hours, 2.4 mg of the capsule remain in your body.

(a)

Time(min) 0 8 16 24 32 10
Temp$(^\circ F)$ 202 188.4 176.2 165.2 155.2 146.4
Temp - 66 136 122.4 110.2 99.2 89.2 80.4

$&\boxed{r=\frac{t_{n+1}}{t_n}=\frac{122.4}{136}=0.9} && \boxed{r=\frac{t_{n+1}}{t_n}=\frac{110.2}{122.4}=0.9} && \boxed{r=\frac{t_{n+1}}{t_n}=\frac{99.2}{110.2}=0.9}\\& \boxed{r=\frac{t_{n+1}}{t_n}=\frac{89.2}{99.2}=0.899} && \boxed{r=\frac{t_{n+1}}{t_n}=\frac{80.4}{89.2}=0.901}$

The common ratio is 0.9. Therefore the dishwasher is cooling at a rate of $\boxed{100\%- 90\%=10\%}$ which is 10% every eight minutes.

(b) The value of ‘$a$’ in $y=a(b)^\frac{x}{c}+d$ is the initial temperature less the temperature in the kitchen. Therefore, the value of ‘$a$’ is $\boxed{136^\circ F}$.

An exponential function to model the temperature of the dishwasher after ‘$t$’ minutes is $\boxed{T=136(0.9)^\frac{t}{8}+66}$

(c) The equation of the horizontal asymptote is $\boxed{y=66}$

(d) The washer’s temperature after 3 hours (180 minutes) is

$& T =136(0.9)^\frac{t}{8}+66 \\& T =136(0.9)^\frac{180}{8}+66 \\& \boxed{T =78.7^\circ F}$

(e) The sketch of the dishwasher’s change in temperature over time is shown below:

Jan 16, 2013

Jan 14, 2015