Question

A library has received a single large contribution of \(\$ 5000 .\) A walkathon is also being held in which each sponsor will contribute \(\$ 10 .\) Let \(x\) represent the number of sponsors. Write a function that represents the average contribution per person, including the single large contributor. Sketch the graph of the function.

Short answer

The function which represents the average contribution per person, including the single large contributor, is \(f(x) = \frac{5000}{x+1} + 10\)

Step-by-step solution

Formulating the Function

The average contribution per person is calculated by dividing the total amount received by the total number of contributors. Here, the total amount received is the sum of the large contribution (\$5000) and the contribution from the sponsors (the number of sponsors 'x' multiplied by the amount each sponsor contributes, which is \$10), which can be expressed as '5000+10x'. The total number of contributors is the number of sponsors 'x' plus one (for the large contributor). Therefore, the function can be formulated as \(f(x) = \frac{(5000+10x)}{(x+1)}\).

Simplifying the Function

The function can be simplified by distributing the division across the addition. So, \(f(x) = \frac{5000}{x+1} + \frac{10x}{x+1}\) simplifies to \(f(x) = \frac{5000}{x+1} + 10\).

Sketching the Graph

To plot the function, first identify important features such as the y-intercept and asymptote. The y-intercept, where x=0, can be found by plugging x=0 into the function, which gives \(5000 +10 = 5010\). This signifies that the initial contribution (when no sponsors are present) is \$5010. Moreover, as 'x' approaches infinity, \(\frac{5000}{x+1}\) should approach 0 and hence the function will approach the value 10, thus indicating the presence of a horizontal asymptote at y = 10. With these characteristics, a graph of the function can be drawn. Notice as the number of sponsors increases, the average contribution is decreasing and tending towards \$10.