3.6 Tools and Strategies for Applying Exponential Models
Chapter
Chapter 3
Section
3.6
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Lectures 2 Videos
Solutions 22 Videos

For each exponential scatter plot, select the corresponding equation of its curve of best fit.

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Q1

Pick one of the unmatched equations from question 1. Sketch a scatter plot that the equation could fit.

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Q2

Toby has invested some money in a mutual fund. The scatter plot shows the value of her investment after the first few years.

Do the data appear to have an exponential trend? Explain your reasoning.

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Q3a

Toby has invested some money in a mutual fund. The scatter plot shows the value of her investment after the first few years.

Estimate values of a and b to develop an exponential model for the data of the form V(n) = a \times b^n. Explain how you arrived at your estimated values.

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Q3b

A fully charged cell phone battery loses 2% of its charge every day.

Determine an equation that models the percent charge, C, that remains in the battery after t days.

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Q4a

A fully charged cell phone battery loses 2% of its charge every day.

Determine the percent charge remaining in the battery after

i) 25 days

ii) 50 days

iii) 75 days

iv) 100 days

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Q4b

A fully charged cell phone battery loses 2% of its charge every day.

Determine the half-life of the battery.

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Q4c

Best Electronics has had constant growth since 1995. Its profits are modelled by the equation P(t) = 7.4(1.59)^t, where P is the yearly profit, in thousands of dollars, and t is the number of years since 1995.

Make a table of values to show annual profits for year 0 (1995) and each of the following 10 years. Approximate values to one decimal place.

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Q5a

Best Electronics has had constant growth since 1995. Its profits are modelled by the equation P(t) = 7.4(1.59)^t, where P is the yearly profit, in thousands of dollars, and t is the number of years since 1995.

Graph the data in the table. Does this represent exponential decay or exponential growth? Explain.

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Q5b

Best Electronics has had constant growth since 1995. Its profits are modelled by the equation P(t) = 7.4(1.59)^t, where P is the yearly profit, in thousands of dollars, and t is the number of years since 1995.

What was the company’s profit in 1995?

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Q5c

Best Electronics has had constant growth since 1995. Its profits are modelled by the equation P(t) = 7.4(1.59)^t, where P is the yearly profit, in thousands of dollars, and t is the number of years since 1995.

Predict the company’s yearly profit in 2015.

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Q5d

Best Electronics has had constant growth since 1995. Its profits are modelled by the equation P(t) = 7.4(1.59)^t, where P is the yearly profit, in thousands of dollars, and t is the number of years since 1995.

According to the model, when does the yearly profit reach $500 000 000?

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Q5e

Air pressure decreases as altitude increases. The following table gives the air pressure, p(a), measured in kilopascals (kPa), at an altitude of a km above sea level.

a) Construct a scatter plot and the curve of best fit for the data.

b) Determine the equation of the exponential function that best represents the data.

c) Determine the air pressure at the summit of each mountain.

i) Mount Logan, altitude 6050 m

ii) Mount Everest, altitude 8848 m

d) What altitude corresponds to an air pressure of 20 kPa?

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Q6

Coffee, tea, cola, and chocolate contain caffeine. When you consume caffeine, the percent, P. of caffeine remaining in your body over time is represented by the function P = 100(0.87)^t, where t is the elapsed time, in hours.

a) Make a table of values for the percent of caffeine remaining for a 24-hour period, in 2-h intervals. Approximate values to one decimal place.

b) Does the function represent exponential growth or exponential decay? Justify your answer graphically.

c) Determine the percent of caffeine remaining in your body after

i) 1 h

ii) 9 h

iii) 15h

d) How long will it take for the percent of caffeine to drop by 50%?

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Q7

In a steel mill, red-hot slabs of steel are pressed many times between heavy rollers. The width of the slab remains the same on every pass; however, the length increases by 20% and the thickness decreases by 17%. Consider a slab that is p metres long and q metres thick.

Write an equation to represent the length, l, in metres, of the slab after n passes.

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Q8a

In a steel mill, red-hot slabs of steel are pressed many times between heavy rollers. The width of the slab remains the same on every pass; however, the length increases by 20% and the thickness decreases by 17%. Consider a slab that is p metres long and q metres thick.

Write an equation to represent the thickness, t, in metres, of the slab after n passes.

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Q8b

In a steel mill, red-hot slabs of steel are pressed many times between heavy rollers. The width of the slab remains the same on every pass; however, the length increases by 20% and the thickness decreases by 17%. Consider a slab that is p metres long and q metres thick.

Use your results from parts a) and b) to write equations for the length and thickness of a slab that is 2.00 m long and 0.50 m thick.

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Q8c

In a steel mill, red-hot slabs of steel are pressed many times between heavy rollers. The width of the slab remains the same on every pass; however, the length increases by 20% and the thickness decreases by 17%. Consider a slab that is p metres long and q metres thick.

How many passes are needed until the length is at least 20 m? How thick is the slab at this point?

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Q8d

In a steel mill, red-hot slabs of steel are pressed many times between heavy rollers. The width of the slab remains the same on every pass; however, the length increases by 20% and the thickness decreases by 17%. Consider a slab that is p metres long and q metres thick.

How many passes are needed until the thickness is about 1 mm? How long is the slab at this point?

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Q8e

Toby has invested some money in a mutual fund. The scatter plot shows the value of her investment after the first few years.

Use the tool of your choice to find an exponential model of these data.

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Q3c

Toby has invested some money in a mutual fund. The scatter plot shows the value of her investment after the first few years.

Use the exponential model you produced in part c) to predict the value of Toby’s investment after 12 years.

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Q3d

Toby has invested some money in a mutual fund. The scatter plot shows the value of her investment after the first few years.

Approximately how long it will take for Toby’s investment to grow to $1000?

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Q3e