Problem 36 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Employers are increasingly turning to GPS (global positioning system) technology to keep track of their fleet vehicles. The estimated number of automatic vehicle trackers installed on fleet vehicles in the United States is approximated by $$ N(t)=0.6 e^{0.17 t} \quad(0 \leq t \leq 5) $$ where \(N(t)\) is measured in millions and \(t\) is measured in years, with \(t=0\) corresponding to 2000 . a. What was the number of automatic vehicle trackers installed in the year \(2000 ?\) How many were projected to be installed in \(2005 ?\) b. Sketch the graph of \(N\).

Step-by-Step Solution

Answer

In the year 2000, there were 0.6 million automatic vehicle trackers installed. In the year 2005, it was projected that 1.4 million trackers would be installed. The graph of the function \(N(t) = 0.6e^{0.17t}\) is an increasing exponential function, with a horizontal asymptote at N=0, starting at the point (0, 0.6) and growing increasingly as t increases, illustrating the growth of installed trackers over time.

Step 1: Number of trackers in the year 2000

To find the number of trackers installed in the year 2000, we need to substitute t = 0 in the given function \(N(t)=0.6 e^{0.17 t}\). We have: $$ N(0) = 0.6 e^{0.17 \cdot 0} $$ Since \(e^0 = 1\), we get: $$ N(0) = 0.6 \cdot 1 = 0.6 $$ So, there were 0.6 million automatic vehicle trackers installed in the year 2000.

Step 2: Number of trackers in the year 2005

To find the number of trackers installed in the year 2005, we need to substitute t = 5 in the given function \(N(t)=0.6 e^{0.17 t}\). We have: $$ N(5) = 0.6 e^{0.17 \cdot 5} $$ Using a calculator, we find the value of \(e^{0.85}\), and then multiply it by 0.6: $$ N(5) = 0.6 \cdot e^{0.85} \approx 0.6 \cdot 2.34 \approx 1.4 $$ So, it was projected that about 1.4 million automatic vehicle trackers would be installed in the year 2005.

Step 3: Sketch the graph of N(t)

To sketch the graph of the function \(N(t) = 0.6e^{0.17t}\), we need to understand the behavior of exponential functions. The given function is of the form \(N(t) = ab^{t}\), where a = 0.6 and b = \(e^{0.17}\). Since b > 1, the graph is an increasing exponential function, which means it will have a horizontal asymptote at N = 0 and it will grow increasingly as t increases. The graph starts at the point (0, 0.6) and approaches infinity as t approaches infinity. To better illustrate this, you can plot the function using graphing software or a graphing calculator, remembering that this function is valid for the domain \(0 \leq t \leq 5\). The increasing nature of the graph's curve will show the growth of the number of automatic vehicle trackers installed in fleet vehicles over time.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent, typically represented as \(a \cdot e^{bt}\). In the exercise, the function given is \(N(t) = 0.6e^{0.17t}\). Here:

• \(a = 0.6\)
• \(b = 0.17\)
• "e" is the natural base of logarithms, approximately equal to 2.718.

An exponential function like this is characterized by its rapid growth rate.
The function increases by a consistent percentage over equal time intervals.
This concept is especially useful in real-world scenarios such as modeling population growth or, in this case, the growth of technology installations like GPS trackers. To understand the function:

• It begins with 0.6 million trackers (when \(t = 0\)), indicating the base amount at the year 2000.
• The exponent \(0.17t\) represents the growth rate over time, generating exponential growth as the years pass.

Graphing Functions

Graphing an exponential function helps us visually interpret the growth or decay process it models. The function \(N(t) = 0.6e^{0.17t}\) represents growth over the years.
To accurately graph this function, keep these steps in mind:

• Identify key points: Since \(N(0) = 0.6\), the graph starts at \((0, 0.6)\) on the y-axis.
• Find another critical point: For \(t = 5\), calculate \(N(5)\) to determine that it equals approximately 1.4 million. Placing this point \((5, 1.4)\) on your graph helps guide the curve.
• Sketch the curve: It's an increasing curve that never touches the x-axis, steadily rising as time progresses.
• Add asymptotes: This function's graph approaches a horizontal asymptote at \(N = 0\), indicating it never actually reaches zero.

Utilizing graphing software or a graphing calculator can offer a precise rendition of this function's curve.
As the graph moves rightward, it visually depicts the accelerating installation of GPS trackers over five years.
The graceful upward slope of the curve is a hallmark of exponential growth.

Mathematical Modeling

Mathematical modeling involves using mathematical language to represent real-world phenomena.
In this exercise, the model is used to predict the number of GPS trackers installed in fleet vehicles over time.From the function \(N(t) = 0.6e^{0.17t}\), you can deduce:

• **Initial Condition:** With \(N(0) = 0.6\), it sets the reality of 0.6 million units installed in 2000.
• **Predictive Ability:** By substituting varying values of \(t\) (like \(t = 5\) for 2005), the function estimates around 1.4 million trackers — showing growth.
• **Real-World Application:** The model aids planning and decision making in business, allowing projections on tech deployment and fleet management.

Mathematical models like this help narrate a story of progression and trends. They simplify complex real-life dynamics into a manageable equation.
This helps stakeholders to understand potential outcomes and strategize effectively.
Therefore, this mathematical model is crucial for anticipating the trajectory of technological installations within a designated timeframe.