Question

Capital Campaign The board of trustees of a college is planning a five-year capital gifts campaign to raise money for the college. The goal is to have an annual gift income \(I\) that is modeled by \(I=2000\left(375+68 t e^{-0.2 t}\right)\) for \(0 \leq t \leq 5\), where \(t\) is the time in years. (T) (a) Use a graphing utility to decide whether the board of trustees expects the gift income to increase or decrease over the five-year period. (b) Find the expected total gift income over the five-year period. (c) Determine the average annual gift income over the five-year period. Compare the result with the income given when \(t=3\).

Short answer

This is just a template response. The actual short answer will be dependent on the exact mathematical results obtained during the solution steps.

Step-by-step solution

Graph the Function

First, graph the function using a graphing tool. This will enable you to visually identify whether the gift income is expected to increase or decrease over the five-year period. Plot the function \(I=2000\left(375+68 t e^{-0.2 t}\right)\), where \(0 \leq t \leq 5\).

Determine the Trend

Analyze the graph obtained in Step 1. If the graph is going upward as \(t\) increases, the gift income is expected to increase over the five-year period. If it's going downward, it's expected to decrease.

Calculate Total Gift Income

To find the expected total gift income over the five-year period, integrate the given function from 0 to 5 as follows: \(\int_0^5 2000\left(375+68 t e^{-0.2 t}\right) dt\). This will give the total expected income as a definite integral.

Find the Average Annual Gift Income

Divide the total gift income (obtained from Step 3) by 5 (the number of years). This will give the average annual gift income.

Compare with Income at t=3

Substitute \(t = 3\) into the original function, \(I=2000\left(375+68 t e^{-0.2 t}\right)\), to obtain the income at three years. Compare this value with the average annual gift income computed in Step 4. If it is less, then the college was earning below average at year 3; if it is more, it was earning above average.

Key concepts

Exponential Decay Function

Exponential decay functions describe processes where quantities decrease over time at a rate proportional to their current value. This concept is crucial in fields like physics, chemistry, and finance. In our context, the annual gift income function given by I = 2000(375 + 68t e^{-0.2t}), where I represents the income and t is the time in years, includes an exponential decay component represented by e^{-0.2t}.

The e here stands for Euler's number, approximately equal to 2.71828, a fundamental mathematical constant. The negative exponent indicates that as time t increases, the value of e^{-0.2t} decreases, reflecting a decline in the additional gift income over time. It's essential to understand this behavior, as it affects the overall income throughout the campaign and sets realistic expectations for future planning.

Integration Applications

Integration serves as a powerful tool to solve diverse problems that require summing infinitesimally small data points to find a whole. When dealing with a capital gifts campaign where income at each instant within a period needs to be summed up, integration is the method of choice.

For instance, determining the total expected gift income over the five years involves finding the definite integral of the income function from 0 to 5. Through integration, we can calculate the area under the curve, which in this case represents the total income from the campaign within the given time frame. Practical applications of this mathematical method range from calculating displacement given a velocity-time graph, to finding the work done by a variable force, to understanding economic models.

Average Value of a Function

The average value of a function over a specific interval gives us the single value that best represents the function's behavior over that interval. It's like finding the 'middle ground' of all the function's values. In a capital gifts campaign, it's useful to know the average income to assess the campaign's consistency and predictability over time.

To find the average annual gift income over five years using the function I = 2000(375 + 68t e^{-0.2t}), integrate over the interval from 0 to 5 and then divide by the interval's length, which is 5 years. This mathematical concept is widely used across sciences and economics, providing vital insights by simplifying complex data into a single representative value.

Mathematical Modeling

Mathematical modeling involves creating equations to represent real-world scenarios. The capital gifts campaign income function is an excellent example of a model that encapsulates both predictable elements and more dynamic aspects affected by time. Modeling allows stakeholders to make informed decisions based on simulations of real-life occurrences.

One of the benefits of mathematical modeling, as seen in this exercise, is the ability to predict future outcomes, like the expected increase or decrease of gift income. By modeling income with exponential functions that take into account factors like time, we can foresee the campaign's performance and structure strategies accordingly. It's the cornerstone of applying mathematical analysis to practical issues and strategic planning.

Definite Integral

A definite integral calculates the net area between the graph of a function and the x-axis over a specified interval. It's fundamental when we want to compute total quantities. In the case of our capital gifts campaign, we're interested in the total gift income, which we find by integrating the income function over the five-year period.

The definite integral considers the contributions of each infinitesimal segment within the range of integration, allowing us to accumulate the total gift income. In other contexts, definite integrals might represent the total mass of a substance distributed along a line, the total charge in an electric field, or the accumulated value of an investment over time. Learning this concept is not just about solving a particular problem but acquiring a versatile tool that can be applied to numerous scenarios.