Problem 85 - Algebra and Trigonometry

Question

A new factory in a small town has an annual payroll of S6 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 will be \$15 million.

Step 1: Identify the first term and the common ratio

In this problem, the first term (\(a\)) is the initial payroll which is \$6 million. The common ratio (\(r\)) is \(60 \%\) or \(0.60\).

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

The formula for the sum (\(S\)) of an infinite geometric series is \(S = \frac{a}{1 - r}\). Plug in the values: \(S = \frac{6,000,000}{1 - 0.60}\).

Step 3: Calculate the total economic impact

Perform the calculation to find the sum of all spending, which is the total economic impact on the town.

Step 4: Interpret the result

The result is the total money that will be spent in the small town each year as a direct or indirect result of the factory's payroll.

Key Concepts

The Economic Impact of a Factory on a Small Town

Imagine a new factory opens up in a small town, providing an annual payroll of $6 million. This influx of money doesn't just benefit the factory workers, but also has rippling effects throughout the entire local economy, a concept known as economic impact. Each dollar paid out by the factory is partially spent within the town, multiplying in effect across various local businesses and services.
Understanding the total economic impact means grasping the full scale of financial benefit the factory brings to the town beyond just salaries. In this exercise, it's important to quantify not just the initial expenditure, but each subsequent spending round resulting from that initial payroll.

• Every round of spending distributes money further into the local economy.
• The initial $6 million payout initiates multiple spending chains.
• This is crucial for assessing the broader economic benefits a single business may provide.

This calculated spending throughout the town illustrates how economic activity can be propagated, showing the complete picture of a factory’s influence on local prosperity.

Understanding the Common Ratio

To determine how the factory's payroll cascades through the town's economy, we must understand the role of the common ratio in a geometric series. The common ratio, denoted as \(r\), represents the fixed proportion of money that gets spent in the town each time money changes hands.
In this scenario, the common ratio is 0.60 or 60%, symbolizing that each dollar a local resident receives from the initial factory payroll and subsequent rounds of spending has a 60% likelihood of being used locally.
Here’s why the common ratio matters:

• It determines the fraction of each spending round that stays within the community.
• A higher common ratio would mean more local retention of funds, amplifying the cumulative economic impact.
• A common ratio of 0.60 indicates a slowing of money retention at each stage, but still significant enough to have a large compounded effect.

Therefore, evaluating and selecting the correct common ratio is key to accurately measuring the spending potential in the town.

Applying the Geometric Series Formula

To calculate the total economic impact, we can use the geometric series formula, which helps determine the sum of an infinite series of spending activities initiated by the factory's payroll.
The formula for an infinite geometric series is:\[S = \frac{a}{1 - r}\]where \(S\) is the sum of the series, \(a\) is the first term (in this case, the $6 million payroll), and \(r\) is the common ratio (0.60).
Applying the values:

• First term, \(a = 6,000,000\).
• Common ratio, \(r = 0.60\).
• Substitute these into the formula:

\[S = \frac{6,000,000}{1 - 0.60}\]This gives us the total sum of spending, reflecting the total economic influence of the factory's presence annually. Breaking it down simplifies understanding how exponential economic impact formulas are derived and applied in real-world scenarios.