Practice Exam 5B

Section 5.7 Practice Exam 5B

Exercises Practice Exam 5B

Try taking this version of the practice exam under testing conditions: no book, no notes, no classmate’s help, no electronics (computer, cell phone, television). Give yourself one hour to work and wait until you have tried your best on all of the problems before checking any answers.

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1.

The number of school children in the district whose first language in not English has been on the rise. The equation describing the situation is

\begin{equation*} C=673(1.043)^T \end{equation*}

where $C$ is the number of school children in the district whose first language is not English, and $T$ is time measured in years (from now).

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(a)

Make a table showing the number of school children in the district whose first language is not English now, in one year, in two years, and in ten years. Don’t forget now too.

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(b)

What percent increase is implicit in this equation?

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(c)

Use successive approximation to determine approximately when there will be over 1,700 school children in the district whose first language is not English. Display your work in a table. Round your answer to the nearest year.

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(d)

Show how to solve the equation to calculate exactly when there will be over 1,700 school children in the district whose first language is not English.

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2.

The lottery jackpot started at $600,000. After 17 days the jackpot had increased to $2.1 million. The lottery is designed so that the jackpot grows exponentially.

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(a)

Name the variables including units.

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(b)

Write an equation describing the jackpot. Hint: find the daily growth factor.

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(c)

By what percentage does the jackpot increase each day?

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(d)

What will the jackpot be after 20 more days (after 37 days total)?

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3.

The creeping vine is taking over Fiona’s front lawn. Write $V$ for the area covered by the vine (in square feet) and $T$ for time in years since she moved into her house.

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(a)

When Fiona moved in, vine covered about 3 square feet. She believes it has doubled each year since. Write an exponential equation showing how the area covered by the vine is a function of time. Stuck? Try making a table first.

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(b)

At some point the vine will take over the entire lawn, so perhaps a saturation model would be better. That equation might be

\begin{equation*} \textbf{Saturation:} \quad V = 170 - 167 \ast 0.8^T \end{equation*}

Another equation would be a logistic model. Perhaps

\begin{equation*} \textbf{Logistic:} \quad V = \frac{129}{1+ 42 \ast 0.34^T} \end{equation*}

Fill in the corresponding rows of the table for each model.

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years	0	1	2	3	4	5	6
area (exponential)	$\fillinmath{\displaystyle\int\int}$	$\fillinmath{\displaystyle\int\int}$	$\fillinmath{\displaystyle\int\int}$	$\fillinmath{\displaystyle\int\int}$	$\fillinmath{\displaystyle\int\int}$	$\fillinmath{\displaystyle\int\int}$	$\fillinmath{\displaystyle\int\int}$
area (saturation)	$\fillinmath{\displaystyle\int\int}$	$\fillinmath{\displaystyle\int\int}$	$\fillinmath{\displaystyle\int\int}$	$\fillinmath{\displaystyle\int\int}$	$\fillinmath{\displaystyle\int\int}$	$\fillinmath{\displaystyle\int\int}$	$\fillinmath{\displaystyle\int\int}$
area (logistic)	$\fillinmath{\displaystyle\int\int}$	$\fillinmath{\displaystyle\int\int}$	$\fillinmath{\displaystyle\int\int}$	$\fillinmath{\displaystyle\int\int}$	$\fillinmath{\displaystyle\int\int}$	$\fillinmath{\displaystyle\int\int}$	$\fillinmath{\displaystyle\int\int}$

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(c)

Draw a graph showing all three models on the same set of axes.

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4.

Many different agencies are working to lower infant mortality. Infant mortality is measured in deaths per thousand births. The world infant mortality rate in 1955 was around 52 (per thousand births). By the year 2000, it was down to around 23.

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(a)

Name the variables, including units and dependence.

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(b)

Write a linear equation modeling infant mortality.

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(c)

Now write an exponential equation modeling infant mortality.

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(d)

Compare the models’ projections for 1955, 1970, 1990, 2000, 2010, and 2020. Summarize your findings in a table.

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(e)

The actual rates were 40 deaths per thousand births in 1970 and 28 deaths per thousand births in 1990. Which model fits this additional data better?

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years	0	1	2	3	4	5	6
area (exponential)	\(\fillinmath{\displaystyle\int\int}\)	\(\fillinmath{\displaystyle\int\int}\)	\(\fillinmath{\displaystyle\int\int}\)	\(\fillinmath{\displaystyle\int\int}\)	\(\fillinmath{\displaystyle\int\int}\)	\(\fillinmath{\displaystyle\int\int}\)	\(\fillinmath{\displaystyle\int\int}\)
area (saturation)	\(\fillinmath{\displaystyle\int\int}\)	\(\fillinmath{\displaystyle\int\int}\)	\(\fillinmath{\displaystyle\int\int}\)	\(\fillinmath{\displaystyle\int\int}\)	\(\fillinmath{\displaystyle\int\int}\)	\(\fillinmath{\displaystyle\int\int}\)	\(\fillinmath{\displaystyle\int\int}\)
area (logistic)	\(\fillinmath{\displaystyle\int\int}\)	\(\fillinmath{\displaystyle\int\int}\)	\(\fillinmath{\displaystyle\int\int}\)	\(\fillinmath{\displaystyle\int\int}\)	\(\fillinmath{\displaystyle\int\int}\)	\(\fillinmath{\displaystyle\int\int}\)	\(\fillinmath{\displaystyle\int\int}\)