Imperial Educational Services (I.E.S.)wants to offer a workshop in pollution control to key personnel at Acme Corporation.I.E.S.will offer the course to thirty persons at a charge of $50 each.Moreover,I.E.S.will agree to reduce the charge for everybody by $1.00 for each person over the thirty who attends,up to a total group size of fifty.It has been determined that the greatest revenue the I.E.S.can receive is $1600.What group size will give this revenue?
A) 34
B) 36
C) 40
D) 44
E) 48

Question

Quizplus · Accepted Answer

Let x be the number of participants over 30 who attend the workshop.

The revenue for the workshop can be expressed as:

R(x) = (30 + x)(50 - x)

where 50 - x represents the reduced charge per person attending.

We want to find the group size that will give us the greatest revenue, which is equivalent to finding the maximum value of R(x).

Expanding R(x), we get:

R(x) = -x^2 + 20x + 1500

Taking the derivative of R(x) with respect to x and setting it equal to 0 to find the critical point(s):

R'(x) = -2x + 20 = 0

x = 10

We can confirm that this is a maximum point by checking that the second derivative of R(x) evaluated at x = 10 is negative (-2).

Therefore, the optimal group size is:

30 + x = 30 + 10 = 40

So the answer is (C) 40.