Problem 44 - Calculus and its applications

Question

The initial weight of a prisoner of war is \(140 \mathrm{lb}\). To protest the conditions of her imprisonment, she begins a fast. Her weight \(t\) days after her last meal is approximated by $$W=140 e^{-0.009 t}$$ a) How much does the prisoner weigh after 25 days? b) At what rate is the prisoner's weight changing after 25 days?

Step-by-Step Solution

Answer

a) After 25 days, her weight is approximately 111.8 lb. b) Her weight is decreasing at a rate of 1.006 lb/day after 25 days.

Step 1: Identify the Given Equation

We are given the weight of the prisoner as a function of time, which is \( W(t) = 140e^{-0.009t} \). This function models how the prisoner's weight decreases over time during a fast.

Step 2: Calculate the Weight after 25 Days

Substitute \( t = 25 \) into the equation \( W(t) = 140e^{-0.009t} \).\[ W(25) = 140e^{-0.009 \times 25} \]Calculate the exponent:\[ -0.009 \times 25 = -0.225 \]Then calculate \( e^{-0.225}\):Using a calculator, \( e^{-0.225} \approx 0.7985 \).Substitute back:\[ W(25) \approx 140 \times 0.7985 \approx 111.8 \text{ lb} \]

Step 3: Differentiate the Weight Function

To find the rate of weight change, differentiate the weight function \( W(t) = 140e^{-0.009t} \).The derivative of \( W(t) \) with respect to \( t \) is found with the chain rule.\[ \frac{dW}{dt} = 140 \times (-0.009)e^{-0.009t} \]Simplify:\[ \frac{dW}{dt} = -1.26e^{-0.009t} \]

Step 4: Calculate the Rate of Weight Change after 25 Days

Substitute \( t = 25 \) into \( \frac{dW}{dt} = -1.26e^{-0.009t} \).\[ \frac{dW}{dt} \bigg|_{t=25} = -1.26e^{-0.009 \times 25} \]From earlier calculation, \( e^{-0.225} \approx 0.7985 \).Substitute back:\[ \frac{dW}{dt} \bigg|_{t=25} = -1.26 \times 0.7985 \approx -1.006 \text{ lb/day} \]

Step 5: Interpret the Results

The interpretation of these results is as follows: After 25 days of fasting, the prisoner weighs approximately 111.8 lb. Moreover, her weight is decreasing at a rate of approximately 1.006 lb per day at that moment in time.

Key Concepts

Weight Loss Model

In this scenario, we are examining a weight loss model, which is a mathematical representation of how an individual's weight changes over time. The model given is an exponential decay function: \( W(t) = 140e^{-0.009t} \). Here, \( W(t) \) symbolizes the weight of the prisoner in pounds, with \( t \) representing the number of days since the start of fasting. The initial weight is established at 140 lb. An exponential decay model is characterized by a rapid drop initially, which gradually slows down over time. This suits the context of fasting where significant weight loss is seen at the beginning, yet it tapers off as the body adjusts. The decay rate in our model is 0.009 per day, indicating a continuous and steady pronounced decrease in weight each day.

Derivative Calculation

The derivative in calculus represents the rate of change of a function with respect to one of its variables. In the context of this problem, we need to find the rate at which the prisoner's weight changes over time.To do this, we calculate the derivative of the weight function \( W(t) = 140e^{-0.009t} \). Applying the chain rule, we differentiate to obtain:\[ \frac{dW}{dt} = 140 (-0.009) e^{-0.009t} = -1.26e^{-0.009t} \]This derivative, \( \frac{dW}{dt} \), gives us the rate at which the weight is changing at any given time \( t \). After 25 days, substituting \( t = 25 \) into this expression allows us to discover the rate of weight loss specifically at that time.

Rate of Change

Rates of change are a fundamental concept in both physics and mathematics, providing insight into how quickly a quantity changes over time.For the prisoner's weight loss problem, the rate of change tells us how quickly her weight is decreasing at a specific time point. By examining the derivative \( \frac{dW}{dt} \), which we previously calculated to be approximately -1.006 lb/day after 25 days, we understand that this is her rate of weight loss at that moment.This rate of change is negative, which highlights that the amount (weight) is decreasing. Additionally, the magnitude of the rate (-1.006 lb/day) helps to clarify the pace at which this change occurs, allowing further insights into the health impacts and persistence of weight loss during prolonged fasting.