Problem 7 - Applied Calculus

Question

Find the doubling time of a quantity that is increasing by \(7 \%\) per year.

Step-by-Step Solution

Answer

10 years.

Step 1: Understand the Formula

To find the doubling time, we can use the Rule of 70. This rule provides a simple way to estimate the doubling time of an investment or quantity that is growing at a constant rate. The formula is: \[ \text{Doubling Time} = \frac{70}{\text{rate in percent}} \] where the rate is the annual growth rate expressed as a percentage.

Step 2: Substitute the Growth Rate

In our problem, the quantity is increasing at a rate of \(7\%\) per year. Substitute this value into the formula: \[ \text{Doubling Time} = \frac{70}{7} \]

Step 3: Perform the Calculation

Divide 70 by 7 to find the doubling time: \[ \text{Doubling Time} = \frac{70}{7} = 10 \] Thus, the doubling time for the quantity is 10 years.

Key Concepts

Exponential Growth

Exponential growth occurs when the increase in a quantity is proportional to its current value, leading to the quantity growing by the same percentage over each equal time period. This type of growth is seen when populations, investments, or other quantities grow at a constant relative rate. Rather than adding a fixed amount each time, the increase is multiplicative, building upon the increased values over time.

Key features of exponential growth include:

• The growth rate is constant in percentage terms, not absolute amounts.
• The larger the quantity, the bigger the increase because the percentage applies to an ever-increasing base.

To visualize exponential growth, imagine a savings account where money accumulates with compound interest. Each year, you earn interest not only on your initial deposit but also on the previously earned interest, leading to rapid growth after a certain period.

Rule of 70

The rule of 70 is a straightforward mathematical formula used to estimate the doubling time of a quantity that grows exponentially. It provides an easy way to understand how long it will take for something like an investment or population to double at a given constant growth rate.

The formula is:

• Doubling Time = \( \frac{70}{\text{growth rate in percent}} \)

For example, if a quantity grows by 7% annually, the rule of 70 helps us quickly calculate that the quantity will double approximately every 10 years by substituting the growth rate into the formula as \( \frac{70}{7} = 10 \).

This rule works well for growth rates that are relatively low and helps provide quick estimates without complex calculations. It's especially useful in economics and demography, where understanding future growth scenarios is crucial.

Growth Rate Calculation

Calculating a growth rate is about finding how much a quantity increases relative to its initial value over a specified period. It is usually expressed as a percentage and is a key component in the rule of 70. Understanding the growth rate is essential for gauging the speed of exponential growth.

Here's how you can calculate the growth rate:

• Identify the initial value and the final value of the quantity.
• Determine the time period over which the change occurs.
• Use the formula: \( \text{Growth Rate} = \left( \frac{\text{Final Value} - \text{Initial Value}}{\text{Initial Value}} \right) \times 100 \).

For ongoing processes, such as annual interest or population changes, the growth rate provides a benchmark to understand how quickly changes are happening. Once known, this percentage can be applied to estimate the doubling time using the rule of 70, or it can be used to forecast future values.