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6.8: Chapter Test

Difficulty Level: At Grade Created by: CK-12
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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 \begin{align*}y\end{align*}-intercept of the graph of \begin{align*}f(x)=4(2)^{x+2}\end{align*}?
    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 \begin{align*}y=4(2)^x-3\end{align*}?
    1. \begin{align*}y=3\end{align*}
    2. \begin{align*}y=0\end{align*}
    3. \begin{align*}y=1\end{align*}
    4. \begin{align*}y=-3\end{align*}
  4. ________ What is the solution for \begin{align*}9.6 \times 10^{-5}-5.4 \times 10^{-6}\end{align*}?
    1. \begin{align*}9.06 \times 10^{-5}\end{align*}
    2. \begin{align*}4.2 \times 10^{-5}\end{align*}
    3. \begin{align*}9.06 \times 10^{-6}\end{align*}
    4. \begin{align*}9.06 \times 10^{-6}\end{align*}
  5. ________ If \begin{align*}f(x)=3^x-2x\end{align*}, what is the value of \begin{align*}f(-1)\end{align*}?
    1. \begin{align*}f(-1)=-\frac{5}{3}\end{align*}
    2. \begin{align*}f(-1)=\frac{7}{3}\end{align*}
    3. \begin{align*}f(-1)=0\end{align*}
    4. \begin{align*}f(-1)=-1\end{align*}
  6. ________ Which one of the following is NOT a growth curve?
    1. \begin{align*}y=(4)^x\end{align*}
    2. \begin{align*}y=(3)^{-x}\end{align*}
    3. \begin{align*}y=\left(\frac{7}{6}\right)^x\end{align*}
    4. \begin{align*}y=\left(\frac{3}{4}\right)^x\end{align*}
  7. ________ What is the solution for the exponential equation \begin{align*}8^{3x-4}+7=71\end{align*}?
    1. \begin{align*}x=0\end{align*}
    2. \begin{align*}x=25\end{align*}
    3. \begin{align*}x=2\end{align*}
    4. \begin{align*}x=20\end{align*}
  8. ________ What is the simplified expression for \begin{align*}x^{-2}+y^{-1}\end{align*}?
    1. \begin{align*}\frac{y+x^2}{x^2y}\end{align*}
    2. \begin{align*}\frac{1}{x^2+y}\end{align*}
    3. \begin{align*}(x+y)^{-3}\end{align*}
    4. \begin{align*}y+x^2\end{align*}
  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 \begin{align*}(6.8 \times 10^4)\times(7.3 \times 10^5)\end{align*}?
    1. \begin{align*}496.4 \times 10^{10}\end{align*}
    2. \begin{align*}4.964 \times 10^{10}\end{align*}
    3. \begin{align*}49.64 \times 10^{10}\end{align*}
    4. \begin{align*}4.964 \times 10^{9}\end{align*}
  11. ________ What is the value of \begin{align*}\left(\frac{1}{2}\right)^{-3}+9^\frac{1}{2}-64^\frac{2}{3}\end{align*}?
    1. -27
    2. 21
    3. -5
    4. 73
  12. ________ Which equation represents the exponential function graphed below?
    1. \begin{align*}y=2(3)^x+4\end{align*}
    2. \begin{align*}y=3(2)^x+4\end{align*}
    3. \begin{align*}y=7(2)^x\end{align*}
    4. \begin{align*}y=4(2)^x+3\end{align*}

Part B

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

  1. Solve the following exponential equations for ‘\begin{align*}x\end{align*}'.
    1. \begin{align*}\sqrt[3]{\frac{9^{x+1}}{27^x}}=81\end{align*}
    2. \begin{align*}(x-2)^\frac{1}{2}=9^\frac{1}{4}\end{align*}
  2. Simplify each of the following:
    1. \begin{align*}\frac{\left(\frac{1}{3}\right)^{-2}+8^\frac{2}{3}+5^{0}}{\left(\frac{1}{49}\right)^\frac{1}{2}}\end{align*}
    2. \begin{align*}\frac{(16^{m-2})(8^{2m+1})}{(4^{x-1})^{-3}}\end{align*}
  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 \begin{align*}66^\circ F\end{align*}.

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

Time(min) 0 8 16 24 32 10
Temp\begin{align*}(^\circ F)\end{align*} 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 ‘\begin{align*}a\end{align*}’ in \begin{align*}y=a(b)^\frac{x}{c}+d\end{align*} and write an exponential function to model the temperature of the dishwasher after ‘\begin{align*}t\end{align*}’ 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.

Answers to Test

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)

\begin{align*}& \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}\end{align*}

(b)

\begin{align*}&(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}\end{align*}

(a)

\begin{align*}&\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} \end{align*}

(b)

\begin{align*}& \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}}\end{align*}

  1. \begin{align*}y &=a(b)^x \\ y &=10(0.70)^4 \\ y &=2.4 \ mg \end{align*}

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

(a)

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

\begin{align*}&\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}\end{align*}

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

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

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

(c) The equation of the horizontal asymptote is \begin{align*}\boxed{y=66}\end{align*}

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

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

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

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CK.MAT.ENG.SE.1.Algebra-I---Honors.6.8
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