Problem 66 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

According to a study conducted in 2003, the total number of U.S. jobs that are projected to leave the country by year \(t\), where \(t=0\) corresponds to the beginning of 2000, is $$ N(t)=0.0018425(t+5)^{2.5} \quad(0 \leq t \leq 15) $$ where \(N(t)\) is measured in millions. How fast was the number of U.S. jobs that were outsourced changing at the beginning of \(2005 ?\) How fast will the number of U.S. jobs that are outsourced be changing at the beginning of \(2010(t=10)\) ? Source: Forrester Research

Step-by-Step Solution

Answer

At the beginning of 2005, the number of U.S. jobs that were outsourced was changing at a rate of approximately \(0.0202\) million per year. At the beginning of 2010, the number of U.S. jobs that were outsourced was changing at a rate of approximately \(0.0372\) million per year.

Step 1: Understanding the Problem

The problem provides a function \(N(t)=0.0018425(t+5)^{2.5}\) to describe the number of jobs that are projected to leave the country for a given year, \(t\). To determine the rate of change, or derivative, of this function is to answer our problem. The value for \(t\) is chosen to align with the years we are interested in: \(t=5\) for the year 2005 and \(t=10\) for the year 2010.

Step 2: Finding the derivative of N(t)

We use the power rule for differentiation: If \(N(x)=a(x+b)^n\), then \(N'(x)=an(x+b)^{n-1}\). Applying this rule to our function, where \(a=0.0018425\), \(b=5\), and \(n=2.5\), we get \[ N'(t) = 0.0018425 * 2.5(t + 5)^{1.5} \]

Step 3: Determine the rate of change for 2005 and 2010

We can plug \(t=5\) into the derivative function to determine how fast jobs were leaving the country in 2005, and \(t=10\) to determine the rate for 2010. For 2005: \[ N'(5) = 0.0018425 * 2.5 * (5 + 5)^{1.5} \approx 0.0202 \text{ million jobs per year} \] For 2010: \[ N'(10)= 0.0018425 * 2.5 * (10 + 5)^{1.5} \approx 0.0372 \text{ million jobs per year} \] At the beginning of 2005, the number of U.S. jobs that were outsourced was changing at a rate of approximately 0.0202 million per year. At the beginning of 2010, the number of U.S. jobs that were outsourced was changing at a rate of approximately 0.0372 million per year.

Key Concepts

Differentiation

Differentiation is an essential concept in calculus, defined as the process of finding a derivative.
The derivative represents the rate at which a function is changing at any given point and is calculated using various rules depending on the function's form.
In the exercise, we use the power rule for differentiation, applicable when differentiating functions of the form \(N(x)=a(x+b)^n\).
This specific rule is straightforward: multiply by the power of the expression and decrease the power by one:

• The constant \(a\) is 0.0018425
• The base value \(b\) is 5
• The power \(n\) is 2.5

Using the power rule, the derivative is:\[N'(t) = 0.0018425 \times 2.5 \times (t + 5)^{1.5}\]This gives us a function that measures the rate of change, helping us understand how quickly the number of outsourced jobs is increasing during specific years.

Rate of Change

The concept of "Rate of Change" involves understanding how quickly one quantity changes with respect to another.
In calculus, the derivative serves as a tool to find this rate of change, telling us how a function is behaving when the independent variable changes incrementally.
For the given exercise, calculating the rate of change helps us quantify the speed at which U.S. jobs were outsourced in different years: 2005 and 2010.
Once we have the derivative function \(N'(t)\), we calculate the rate for each specific year by substituting \(t\) into the derivative.

• For 2005, \(t=5\), the rate is approximately 0.0202 million jobs per year
• For 2010, \(t=10\), the rate is approximately 0.0372 million jobs per year

These rates define how rapidly job outsourcing was changing during those years, providing insights into economic and labor trends.

Mathematical Modeling

Mathematical modeling involves using mathematical expressions to represent real-world scenarios.
By forming equations or inequalities, we can describe, predict, and analyze various phenomena.
In this exercise, the function \(N(t) = 0.0018425(t+5)^{2.5}\) is a model designed to estimate the outsourcing of U.S. jobs over time.
It is based on factors such as historical data and future expectations. Mathematical models like this one allow economists, business leaders, and policy makers to visualize potential future trends and make informed decisions.

• Models are simplified representations, helping to break down complex real-world behaviors
• They provide a way to conduct "what if" scenarios, like predicting job outsourcing in future years

When using such models, understanding their parameters (such as \(a\), \(b\), and \(n\) in our function) helps professionals adapt predictions to fresh data, ensuring that their forecasts stay relevant.