APPLICATIONS OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS

APPLICATIONS OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS

(Population Word Problems)

To solve an exponential or logarithmic word problem, convert the narrative to an equation and solve the equation.

Now we are going to get serious.

Example 19: According to the U.S. Bureau of the Census, in the year 1850 the population of the United States was 23,197,876; in 1900, the population was 62,947,714.

(A) Assuming that the population grew exponentially during this period, find the model (equation) that describes the population growth. What is the relative growth rate?

(B) Assuming continued growth at the same rate, predict the 1950 population.

(C) The actual population for 1950, according to the Bureau of the Census, was 150,697,361. How does this compare to your prediction in part (B)? Was the actual growth over the period 1900-1950 faster or slower than the exponential growth with the growth rate in part (A)?

Solution and Explanations:

The problem states that the growth rate was exponential during the years 1850 through 1900. You many use any positive number for the base in the exponential equation (model). However, for the sake of standardization, let's choose the base e. The generic equation is

where a is the population when t = 0 and b is the relative growth rate with respect to the base e. Since the study started in 1850, we will let t = 0 in the year 1850.

• Let t = 0 and f(t) = 23, 197,876 in the equation .

  

• Rewrite the equation with a = 23,197,876.

  

• We know that in 1900 (when t = 50), the population was 62,947,714. Substitute 62,947,714 for f(t) and 50 for t in the equation

  

• Divide both sides by 23,197,876.

  

• Take the natural logarithm of both sides of the equation;

  

• Simplify the right side of the equation using the third rule of logarithms.

  

• Divide both sides by 50 and simplify.

  

  rounded to .02. The equation is

  

  The relative growth rate (with respect to the base e) between the years 1850 and 1900 was approximately 2%.

Predict the 1950 population using the model .

• Substitute t = 100=(1950 - 1850) in the above equation.

  

The actual population was 150,697,361 in 1950. Was the actual growth rate between 1900 and 1950 faster or slower than 2%?

• Remember the 2% is the relative growth rate between 1850 and 1900. To find the exponential growth rate between 1900 and 1950, we have to recalculate that rate. In this study, 62,947,714 is the starting population and 150,697,361 is the population when t = 50.

• Substitute t = 50 and f(t) = 150,697,36 in the equation .

  

• Divide both sides by 62,947,714.

  

• Take the natural logarithm of both sides of the equation;

  

• Simplify the right side of the equation using the third rule of logarithms.

  

• Divide both sides by 50 and simplify.

  

  rounded to .018.

The equation is

The relative growth rate (with respect to the base e) between the years 1900 and 1950 was approximately 1.8%, a little less than the relative growth rate of 2% between the years 1850 and 1900.

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