Problem 45 - Calculus and its applications

Question

The number of women earning a bachelor's degree from a 4 -yr college in the United States grew from 48,869 in 1930 to approximately 920,000 in 2010. (Source: National Center for Education Statistics.) Find an exponential function that fits the data, and the exponential growth rate, rounded to the nearest hundredth of a percent.

Step-by-Step Solution

Answer

The exponential growth function is \( N(t) = 48869 \cdot e^{0.070625t} \) with a growth rate of 7.06% per year.

Step 1: Understanding the General Form of Exponential Functions

An exponential function modeling a situation can typically be expressed as \( N(t) = N_0 \cdot e^{rt} \), where \( N(t) \) is the future value, \( N_0 \) is the initial value, \( e \) is Euler's number, \( r \) is the growth rate, and \( t \) is the time in years. Our task is to find \( N_0 \) and \( r \) given the data.

Step 2: Assign Known Values

From the problem, in 1930, \( N_0 = 48869 \). In 2010 (80 years later), \( N(80) = 920000 \). We'll use these values to find the exponential function parameters.

Step 3: Solve for the Growth Rate r

Using the exponential function formula \( N(t) = N_0 \cdot e^{rt} \), substitute the known values: \( 920000 = 48869 \cdot e^{80r} \). To find \( r \), divide both sides by 48869, obtaining \( e^{80r} = \frac{920000}{48869} \).

Step 4: Simplify and Solve for r

Calculate \( \frac{920000}{48869} \approx 18.829 \). Then take the natural logarithm of both sides: \( 80r = \ln(18.829) \). Solve for \( r \) to find \( r = \frac{\ln(18.829)}{80} \approx 0.070625 \).

Step 5: Convert r to Percentage

Multiply \( r \) by 100 to convert it to a percentage: \( r \approx 0.070625 \times 100 = 7.063 \). Therefore, the growth rate is approximately \( 7.06\% \), rounded to the nearest hundredth percent.

Key Concepts

Exponential Functions

When we talk about exponential functions, we refer to mathematical expressions that model quantities growing or shrinking rapidly. In our scenario, we are looking at the number of women earning bachelor's degrees, which has grown significantly over time. An exponential function typically has the form \( N(t) = N_0 \cdot e^{rt} \), where:

• \( N(t) \) is the quantity at time \( t \).
• \( N_0 \) is the initial quantity (the base number from which growth starts).
• \( e \) is Euler's number, approximately equal to 2.718, a fundamental constant in mathematics.
• \( r \) is the growth rate, determining how fast the quantity grows or decays.
• \( t \) stands for time, often in years.

Let's delve deeper: In 1930, there were 48,869 women graduating, and this number is our \( N_0 \). By 2010, this became 920,000. Our task was to find how quickly this number grew, needing us to solve for \( r \).
This model helps efficiently predict the future values of growing quantities - ideal for educational statistics as they track changes over generations.

Growth Rate Calculation

Understanding growth rates is essential, especially in examining long-term trends such as educational attainment over decades. In the exponential growth scenario, the growth rate \( r \) plays a pivotal role. Here, we found \( r \) using some mathematical steps:
First, you set up the equation: \( N(t) = N_0 \cdot e^{rt} \). For our example in 1930, \( N_0 = 48869 \) and in 2010, which is 80 years later, \( N(80) = 920000 \). Replacing these values, our equation turns into \( 920000 = 48869 \cdot e^{80r} \).
To isolate \( r \), simplify the equation:

• Divide both sides by 48869, obtaining \( e^{80r} = \frac{920000}{48869} \).
• Approximate this division to get around 18.829.
• Then take the natural logarithm of both sides, making it easier to extract \( r \).

Finally, solve for \( r \): \( 80r = \ln(18.829) \)
From this, \( r \approx \frac{\ln(18.829)}{80} \approx 0.070625 \).
Turn \( r \) into a percentage by multiplying by 100: \( r \approx 7.06\% \). This percentage symbolizes how rapidly our initial count escalated to its 2010 value.

Educational Statistics

Educational statistics give us a window into changing educational trends and achievements over time. By using data from sources such as the National Center for Education Statistics, we get valuable insights into how educational attainment evolves throughout decades.
In our example, from 1930 to 2010, the number of women receiving bachelor’s degrees grew exponentially. This shift wasn't gradual— it was a sign of increased access to higher education for women and societal changes supporting female education.
Using exponential functions in educational statistics helps us:

• Predict future trends and make informed decisions about education policies.
• Understand historical shifts and why they occurred.
• Identify areas needing attention, such as programs that support female education.

Through proper analysis and understanding of these numbers, policymakers and educators can craft strategies that continue to promote equality and educational opportunities for all.