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6.8: Chapter Test

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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 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.

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)

& \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:

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Date Created:

Jan 16, 2013

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Jun 04, 2014
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