Question

A student who wishes to use a paper cutter at a local library must buy a membership. The library charges $10 for membership. Sixty students purchase the membership. The library estimates that for every $1 increase in the membership fee, 5 fewer students will become members. What membership fee will provide the maximum revenue to the library?

Answer 1

$31

Explanation:

Let x be the number of dollars of the membership fee. Then, the number of students who will become members is:

60 - 5(x - 10)

This expression comes from the given estimate that for every $1 increase in the membership fee, 5 fewer students will become members. When the fee is $10, 60 students become members, so we need to subtract 5 for every dollar above $10.

The revenue earned by the library is the product of the membership fee and the number of students who become members:

R = x(60 - 5(x - 10)) = 60x - 5x^2 + 250x - 1500

Simplifying this expression, we get:

R = -5x^2 + 310x - 1500

This is a quadratic function with a negative coefficient for the x^2 term, which means it is a downward-facing parabola. Therefore, the maximum revenue occurs at the vertex of the parabola.

The x-coordinate of the vertex can be found using the formula:

x = -b/(2a)

where a is the coefficient of the x^2 term and b is the coefficient of the x term. In this case, a = -5 and b = 310, so:

x = -310/(2*(-5)) = 31

Therefore, the membership fee that will provide the maximum revenue to the library is $31.