Problem 37 - Calculus Concepts

Question

Jobs The rate of change in the number of jobs for a Michigan roofing company is increasing by approximately 6.14 jobs per month squared. The number of jobs in January is decreasing at the rate of 0.87 job per month, and company records indicate that the company had 14 roofing jobs in February. a. Write a differential equation for the rate at which the rate of change in the number of roofing jobs for this company is changing. b. Find a particular solution to the differential equation in part \(a\). c. Use the result of part \(b\) to estimate the number of roofing jobs in August and the number in November.

Step-by-Step Solution

Answer

a) \(\frac{d^2J}{dt^2} = 6.14\); b) \(J(t) = 3.07t^2 + 5.27t + 14\); c) 172 jobs in August, 310 jobs in November.

Step 1: Understand the Variables

Let \( J(t) \) be the number of jobs at time \( t \), where \( t \) is measured in months with \( t = 0 \) representing February. The rate of change of jobs, \( \frac{dJ}{dt} \), is given initially. We know the second derivative, \( \frac{d^2J}{dt^2} = 6.14 \), represents the acceleration or the rate at which the rate of change of jobs changes.

Step 2: Write the Differential Equation

From the information given, we know the second derivative of the number of jobs with respect to time. Therefore, the differential equation representing the situation is:\[ \frac{d^2J}{dt^2} = 6.14 \]

Step 3: Integrate the Differential Equation

First, integrate the differential equation \( \frac{d^2J}{dt^2} = 6.14 \) to find \( \frac{dJ}{dt} \). This gives:\[ \frac{dJ}{dt} = 6.14t + C \]where \( C \) is a constant of integration.

Step 4: Use Initial Condition to Find C

Given that the rate of change of jobs in January (\( t = -1 \)) was decreasing at 0.87 job per month, we use:\[ \frac{dJ}{dt} = -0.87 \] when \( t = -1 \).Substitute to find \( C \):\[ -0.87 = 6.14(-1) + C \]\[ -0.87 = -6.14 + C \]\[ C = 5.27 \]

Step 5: Integrate Again to Find J(t)

Now, integrate \( \frac{dJ}{dt} = 6.14t + 5.27 \) with respect to \( t \) to find \( J(t) \):\[ J(t) = 3.07t^2 + 5.27t + D \]where \( D \) is another constant of integration.

Step 6: Use Another Initial Condition to Find D

Given that there were 14 jobs in February (\( t = 0 \)), we use:\[ J(0) = 14 \]Substitute to find \( D \):\[ 14 = 3.07(0)^2 + 5.27(0) + D \]\[ D = 14 \]

Step 7: Particular Solution for J(t)

Combining everything, the particular solution for \( J(t) \) is:\[ J(t) = 3.07t^2 + 5.27t + 14 \]

Step 8: Estimate Jobs in August

August is 6 months after February, so \( t = 6 \). Substitute into \( J(t) \):\[ J(6) = 3.07(6)^2 + 5.27(6) + 14 \]Calculate:\[ J(6) = 3.07 \times 36 + 31.62 + 14 = 126.12 + 31.62 + 14 = 171.74 \]Therefore, there are approximately 172 jobs in August.

Step 9: Estimate Jobs in November

November is 9 months after February, so \( t = 9 \). Substitute into \( J(t) \):\[ J(9) = 3.07(9)^2 + 5.27(9) + 14 \]Calculate:\[ J(9) = 3.07 \times 81 + 47.43 + 14 = 248.67 + 47.43 + 14 = 310.1 \]Therefore, there are approximately 310 jobs in November.

Key Concepts

Rate of Change

The rate of change is a measure that tells us how a quantity changes with respect to another variable. For example, in the context of a roofing company, it represents how the number of jobs changes over time. Here, we are dealing with a situation where the change is not constant but influenced by acceleration. This is reflected in the second derivative of a function. The second derivative tells us how the rate of change itself is changing. For this exercise, the second derivative is given as a constant, meaning the acceleration rate is steady.

Understanding this concept is vital in modeling real-life situations, such as estimating future job numbers. The first derivative, often denoted as \( \frac{dJ}{dt} \), shows how the number of jobs \( J \) changes over time \( t \). If \( \frac{dJ}{dt} \) is increasing, more jobs are added over time, and vice versa.

• First derivative \( \frac{dJ}{dt} \): Indicates how fast the job numbers are changing.
• Second derivative \( \frac{d^2J}{dt^2} \): Indicates how the change rate itself is changing, i.e., the acceleration.

Integration

Integration is a mathematical process used to find functions when we know their derivatives. In this problem, integration allows us to find the function \( J(t) \), representing the number of jobs at any given time.

Since the second derivative \( \frac{d^2J}{dt^2} = 6.14 \) is known, integrating this provides the first derivative \( \frac{dJ}{dt} \). Through integration, we calculate:

• \( \frac{dJ}{dt} = 6.14t + C \): Integrating the second derivative, introducing an integration constant \( C \) because the integral of a constant is a linear function.
• \( J(t)= 3.07t^2 + 5.27t + D \): Further integration gives \( J(t) \), where \( D \) is another constant from integration.

Integration helps transition from known rates of change to the actual quantities, assisting in predicting real-world scenarios effectively.

Initial Conditions

Initial conditions are crucial in solving differential equations as they allow us to determine constants introduced during integration. In this problem, we were given:

• The rate of change of jobs in January \( (t = -1) \) was \(-0.87\) jobs per month. This condition helps find the constant \( C \).
• There were 14 jobs in February \( (t = 0) \). This condition helps determine the constant \( D \) in the function \( J(t) \).

Applying these conditions to the integrated functions:

• \( C \) was calculated as \( 5.27 \), balancing the initial rate given in January.
• \( D \) was found to be \( 14 \), ensuring that the number of jobs in February started correctly in the solution \( J(t) \).

These initial values anchor our solution in reality, allowing us to provide accurate predictions for future points.

Time-Dependent Functions

Time-dependent functions model how quantities evolve over time, such as job numbers in this case. These functions are crucial for analyzing and predicting future trends in various fields.

In our example, the function \( J(t) = 3.07t^2 + 5.27t + 14 \) was derived to represent the number of roofing jobs at any time \( t \). This quadratic function provides insights into the expected increase or decrease in jobs over the months, grounded in both the rate of change and initial conditions.

Using time-dependent functions:

• Estimate future values, like the job count in August \( t=6 \) and November \( t=9 \).
• Analyze trends, determining if the job market is growing or declining.
• Plan resources accordingly based on anticipated needs.

This approach turns complex data into usable forecasts, essential for strategic planning and decision-making.