Problem 38 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The percentage of families that were married households between 1970 and 2000 is approximately $$ P(t)=86.9 e^{-0.05 t} \quad(0 \leq t \leq 3) $$ where \(t\) is measured in decades, with \(t=0\) corresponding to the beginning of 1970 . a. What percentage of families were married households at the beginning of \(1970,1980,1990\), and 2000 ? b. Sketch the graph of \(P\).

Step-by-Step Solution

Answer

In 1970, 86.9% of families were married households. In 1980, approximately 82.66% of families were married households. In 1990, approximately 78.75% of families were married households. In 2000, approximately 75.16% of families were married households. The graph of \(P(t)=86.9 e^{-0.05 t}\) represents an exponential decay function, starting at 86.9% and decreasing over time.

Step 1: Calculate the percentage for each year

The function representing the percentage of married households over time is given by \(P(t)=86.9 e^{-0.05 t}\). To find the percentage at different years, substitute the corresponding \(t\) values for each year. a. - For 1970, at the beginning, \(t = 0\). So, \[ P(0) = 86.9 e^{-0.05 \times 0} = 86.9 e^0 = 86.9 \] Thus, in 1970, 86.9% of families were married households. - For 1980, \(t = 1\). So, \[ P(1) = 86.9 e^{-0.05 \times 1} = 86.9 e^{-0.05} \approx 82.66 \] Thus, in 1980, approximately 82.66% of families were married households. - For 1990, \(t = 2\). So, \[ P(2) = 86.9 e^{-0.05 \times 2} = 86.9 e^{-0.1} \approx 78.75 \] Thus, in 1990, approximately 78.75% of families were married households. - For 2000, \(t = 3\). So, \[ P(3) = 86.9 e^{-0.05 \times 3} = 86.9 e^{-0.15} \approx 75.16 \] Thus, in 2000, approximately 75.16% of families were married households.

Step 2: Sketch the graph of P(t)

The function \(P(t) = 86.9 e^{-0.05t}\) is an exponential decay function. It starts at a high value around \(86.9\%\) and decreases as the time (t in decades) increases. The graph of P(t) would look like a downward-sloping curve that never reaches 0 since it approaches a horizontal asymptote at \(P=0\). To sketch the graph, plot the points we calculated in step 1: (0, 86.9), (1, 82.66), (2, 78.75), and (3, 75.16). Then, draw a curved line that starts from (0, 86.9) and passes through the other three points to represent P(t).

Key Concepts

Percentage Calculation

When dealing with exponential decay, calculating percentages at specific points in time is essential. This helps understand how a quantity decreases over time. In our exercise, the quantity being observed is the percentage of married households from 1970 to 2000. Here, the decay is represented by the formula \( P(t) = 86.9 e^{-0.05t} \). It's an exponential function where:

• \( P(t) \) is the percentage of married households
• \( e \) is the base of natural logarithms (\( e \approx 2.718 \))
• \( t \) is the time in decades starting from 1970
• \(-0.05 \) is the decay rate

By substituting values of \( t \) (e.g., 0 for 1970, 1 for 1980), we can calculate the exact percentage of married households at each decade. This method highlights how exponential decay can be applied to real-world scenarios, showing a gradual reduction over time.

Graph Sketching

Sketching the graph of an exponential decay function helps visualize trends over time. For our function \( P(t) = 86.9 e^{-0.05t} \), the graph is expected to decrease steadily because of the negative exponent.To sketch:

• Start by plotting known data points calculated earlier: (0, 86.9%), (1, 82.66%), (2, 78.75%), and (3, 75.16%). These points represent the percentage of married households at the beginning of each decade from 1970 to 2000.
• Connect these points with a smooth curve that starts at the highest value and gently slopes down, reflecting the decrease.
• Remember, this curve approaches, but never reaches, the horizontal line at \( P = 0 \), known as an asymptote. This indicates that while the percentage decreases, it doesn't drop to zero within the chart's range.

Graph sketching transforms numerical data into a visual format that makes interpreting and understanding the decline easily accessible. This is particularly useful in applied mathematics to draw insights at a glance.

Applied Mathematics

Applied mathematics often involves using mathematical models, like the exponential decay formula, to solve real-life problems. In this exercise, we modeled the decline in married households from 1970 to 2000. This application entails:

• Identifying the function type (exponential decay here) that best represents the data behavior.
• Utilizing the mathematical model to predict or analyze changes over time. We used the function to compute specific percentages for different decades.
• Using insights from the model to understand broader social or demographic trends, such as how societal norms regarding marriage have evolved over these three decades.

Applied mathematics empowers us to process data through precise lenses, transforming raw numerical information into meaningful narratives that support decisions and policy-making in varied fields.