Problem 85 - College Algebra

Question

Use the formula for the sum of an infinite geometric series to solve. A new factory in a small town has an annual payroll of \(\$ 6\) million. It is expected that \(60 \%\) of this money will be spent in the town by factory personnel. The people in the town who receive this money are expected to spend \(60 \%\) of what they receive in the town, and so on. What is the total of all this spending, called the total economic impact of the factory, on the town each year?

Step-by-Step Solution

Answer

The total economic impact of the factory on the town each year is \$15 million.

Step 1: Identify the first term and the common ratio

In this case, the first term (a) in the geometric series is the initial amount of money on the payroll, which is \$6 million. The common ratio (r) is the percentage of the money that is spent in the town each time, which is 60% or 0.6.

Step 2: Apply the formula for the sum of an infinite geometric series

Now, substitute the values of a and r into the formula S = a / (1 - r). This gives S = \$6 / (1 - 0.6) million.

Step 3: Calculate the total sum

Performing the calculation in the formula gives S = \$6 / 0.4 = \$15 million. This is the total economic impact of the factory on the town each year.