Problem 67 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

WoRKING MoTHERS The percentage of mothers who work outside the home and have children younger than age \(6 \mathrm{yr}\) is approximated by the function $$ P(t)=33.55(t+5)^{0.205} \quad(0 \leq t \leq 21) $$ where \(t\) is measured in years, with \(t=0\) corresponding to the beginning of \(1980 .\) Compute \(P^{\prime}(t) .\) At what rate was the percentage of these mothers changing at the beginning of \(2000 ?\) What was the percentage of these mothers at the beginning of \(2000 ?\)

Step-by-Step Solution

Answer

At the beginning of 2000, the percentage of working mothers with children below the age of 6 was approximately 65.14%, and it was increasing at a rate of 0.86% per year.

Step 1: Deriving the function

We start by differentiating the function \(P(t) = 33.55(t+5)^{0.205}\) with respect to \(t\). Using the power rule, \(P'(t) = 33.55 * 0.205*(t+5)^{0.205-1}\). Simplify this to: \(P'(t) = 6.87175*(t+5)^{-0.795}\).

Step 2: Evaluating the derivative at the specific point

With \(P'(t)\) found, we now need to evaluate \(P'(20)\) which corresponds to the rate at which the percentage of working mothers was changing at the start of 2000. So, \(P'(20) = 6.87175*(20+5)^{-0.795} = 0.86\). This indicates that the percentage of mothers working outside the home was increasing at a rate of 0.86% per year at the beginning of 2000.

Step 3: Evaluating the original function at the specific point

Lastly, we evaluate \(P(t)\) at \(t = 20\) to find the percentage of working mothers at the start of 2000. So, \(P(20) = 33.55*(20+5)^{0.205} = 65.14\). So, the percentage of mothers with children less than 6 years who were working outside the home at the start of the year 2000 was approximately 65.14%.

Key Concepts

Derivative of a Function

The derivative of a function is a foundational concept in calculus representing the instantaneous rate of change of the function with respect to one of its variables. Imagine driving a car where your speedometer shows how quickly you're going at any exact second. The derivative is like that speedometer, but instead of tracking speed, it's measuring how one quantity changes as another changes.

For example, if you have a function that describes how the temperature changes over time, the derivative will tell you how rapidly the temperature is rising or falling at any particular moment. This is exactly what we're looking at when we differentiate the function that represents the percentage of working mothers.

Using calculus, we can find the derivative, denoted as \(P'(t)\), by applying specific rules like the power rule to the function \(P(t)\). This process gives us a new function that provides the rate of change at any given point in time, which can be incredibly valuable in both theoretical and real-world applications.

Power Rule Differentiation

The power rule is one of the simplest and most used rules for differentiating functions in calculus. It states that if you have a function of the form \( x^n \), where \(n\) is any real-number exponent, the derivative of that function is \( n*x^{n-1} \).

When working with the given function \( P(t) = 33.55(t+5)^{0.205} \), we apply this rule to take the derivative. You simply multiply the exponent by the coefficient and decrease the exponent by one. It's a quick and efficient way to understand how a function's output changes in response to changes in the input.

In our example, the power rule was employed to differentiate \( (t+5)^{0.205} \) resulting in \( 0.205*(t+5)^{-0.795} \) (since \( 0.205 - 1 = -0.795 \) ). Multiplying by the coefficient \(33.55\) gives us the derivative \( P'(t) \).

Rate of Change

Understanding the rate of change is critical for interpreting the behavior of functions. It essentially tells us how fast or slow a quantity represented by a function is changing at a particular instant. In practical terms, it helps in predicting trends and making decisions based on those predictions.

For the exercise we're considering, the rate of change is not about speed or temperature—it's about the percentage of working mothers. By evaluating the derivative at \( t = 20 \), which corresponds to the year 2000, we find that the percentage of working mothers was increasing at a rate of 0.86% per year. This rate provides insight into societal trends at that point in time. Moreover, calculating this rate is not only a matter of plugging numbers into the derivative; it is also about understanding how the lives of people—and in this case, working mothers—are evolving over time.