Applications of Exponential and Logarithmic Functions



Applications of Exponential and Logarithmic Functions

1) The cost of tuition for at four-year public universities has been increasing roughly exponentially for the past 5 years. In 1996, the average cost of tuition was $2,975 per year. By 1997, the figure had risen to $3,211. A. Find the growth factor, b, and explain what it tells you about tuition increases. B. Find the equation that represents the average cost of tuition at four-year universities as a function of time, with [pic]corresponding to 1996. C. Use this model to estimate the average cost of tuition in 2003. D. Find the year that the average tuition would be $10,000.

Solution: A. [pic] so 1.07933 is growth factor and this tells you that the tuition is rising by 7.933% each year. B. [pic] C. [pic]

D. [pic]. So in a little less than 16 years (2011) the tuition will be $10,000.

2) A radioactive substance has a half-life of 10 years. Find the equation that describes the quantity of the substance present at time t using the continuous exponential model. Explain what k tells you. What percentage of the original amount would remain after 22 years?

Solution: [pic]. The [pic] tells you that the quantity is decreasing at a continuous annual rate of 6.9315 each year.

[pic] which is 21.7636%. So 21.7636% of the original amount remains after 22 years

3) Suppose 80 ounces of a radioactive substance decays to 9 ounces in 9 hours. What is the half-life of the substance?

Solution: [pic]

[pic]

So the half-life is approximately 2.86 hours

4) Atmospheric pressure is related to height above sea level according to an exponential model. Suppose the pressure at 18,000 feet is half that at sea level. Find the value of k in the continuous model and explain what the value of k tells you. Find the equation and use it to estimate the pressure at 1000 feet, as a percentage of the pressure at sea level.

Solution: [pic]

So [pic]tells you that the atmospheric pressure is dropping by a continuous rate of 0.00385% per foot.

Equation: [pic] [pic]. So the percentage of the pressure at sea level is 96.223%

5) A certain lake is stocked with 1000 fish. The population is growing according to the logistics curve:[pic] where t is measured in months since the lake was initially stocked.

A. Find the population in 8 months.

[pic] fish

B. After how many months will the fish population be 2000?

[pic]

so [pic] months

C. Is there a maximum possible fish population that the lake can sustain? What graphical feature is this on the graph?

The maximum possible fish population is 10,000 fish ([pic]) which is the horizontal asymptote on the graph.

6) A. Let [pic] and [pic]represent the decibel ratings of sounds of intensity [pic] and [pic]respectively. Using log properties, find a simplified formula for the difference between the two ratings, [pic], in terms of the two intensities, [pic] and [pic]. (Decibels are introduced in the text on page 163.)

Solution: [pic]

B. If a sound’s intensity doubles, how many decibels louder does the sound become?

Solution: [pic] so [pic]. So of sound intensity doubles, the decibel rating goes up by approximately 3 dB.

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