Problem 39 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

EDUCATIONAL FUNDING A private college in the southwest has launched a fund- raising campaign. Suppose that college officials estimate that it will take \(f(x)=\frac{10 x}{150-x}\) weeks to reach \(x \%\) of their goal. a. Sketch the relevant portion of the graph of this function. b. How long will it take to reach \(50 \%\) of the campaign's goal? c. How long will it take to reach \(100 \%\) of the goal?

Step-by-Step Solution

Answer

It takes 5 weeks to reach 50% and 20 weeks to reach 100% of the goal.

Step 1: Understanding the function

The function given is \[ f(x) = \frac{10x}{150-x} \]. It represents the number of weeks it will take to reach a certain percentage \(x\%\) of the fundraising goal.

Step 2: Sketching the graph

To sketch the graph, consider values of \(x\) within the relevant domain. Note that as \(x\) approaches 150, the denominator approaches zero, making the function undefined at \(x = 150\). Plot several points within the range \(0 < x < 150\) to visualize the function's behavior. For example, plot points for \(x = 10, 25, 50, 75\). Observe that the function increases and becomes very steep as \(x\) nears 150.

Step 3: Calculating time for 50% goal

To find out how long it will take to reach 50% of the goal, substitute \(x = 50\) into the function: \[ f(50) = \frac{10 \cdot 50}{150 - 50} = \frac{500}{100} = 5 \] weeks.

Step 4: Calculating time for 100% goal

To determine the time to reach 100% of the goal, substitute \(x = 100\) into the function: \[ f(100) = \frac{10 \cdot 100}{150 - 100} = \frac{1000}{50} = 20 \] weeks.

Step 5: Conclusion

Summarize the results from the calculations and the graph. The graph should show that the time to reach higher percentages of the goal increases rapidly as \(x\) approaches 150.

Key Concepts

Fundraising Model

In this exercise, the fundraising model provided by the function helps the college estimate when they will reach certain percentages of their fundraising goal. The function given is:
\[ f(x) = \frac{10x}{150-x} \]
Here, x represents the percentage of the goal that needs to be reached, and f(x) represents the number of weeks it will take.
Understanding this model is key to predicting timelines and planning the campaign effectively. By analyzing how x affects f(x), we can help the college manage resources and set realistic expectations for their fundraising progress.

Function Analysis

Function analysis involves examining the characteristics and behavior of a given function. For the fundraising model, we need to understand the different aspects of the function
\[ f(x) = \frac{10x}{150-x} \].
Key points include:

• The function is undefined at x = 150 since the denominator becomes zero.
• As x approaches 150, the value of f(x) increases sharply, meaning the time to reach higher percentages grows very quickly.
• To find specific values such as f(50) and f(100), we substitute these values into the function and solve.

Analyzing these factors helps us create a clearer picture of the fundraising timeline.

Graphing Rational Functions

When graphing rational functions like
\[ f(x) = \frac{10x}{150-x} \],
it's important to:

• Identify points where the function is undefined (here, at x = 150).
• Calculate and plot several points to understand the function’s behavior within its domain.

For example, plotting points for x = 10, 25, 50, and 75 will show the curve’s shape. The graph demonstrates that the time needed increases faster as the percentage goal x approaches 150, showing a steep upwards trend.

Domain and Range

The domain and range of a function tell us about the possible inputs and outputs. For the given function
\[ f(x) = \frac{10x}{150-x} \],
we have:

• Domain: All real numbers x such that 0 < x < 150. This means x must be between 0 and 150, excluding these endpoints.
• Range: All possible values of f(x). As x approaches 150, f(x) tends towards infinity. This means the range is all positive real numbers.

Knowing the domain and range helps us understand the limitations and expectations of the fundraising timeline.

Problem-Solving Steps

To solve problems using this fundraising model, follow these steps:

• Step 1: Understand the function. Identify the components and what they represent.
• Step 2: Sketch the graph. Use several points to visualize how the function behaves.
• Step 3: Calculate specific values. Substitute particular percentages into the function to find the weeks required.
• Step 4: Interpret results. Summarize the graph and calculations to draw conclusions about the fundraising timeline.
• Step 5: Apply the insights. Use the findings to assist in planning and managing the fundraising campaign.

These steps help break down problems into manageable parts, aiding in a deeper understanding and easier application of the concepts.