Mathematical model problem...?

Question

A travel agency offers an organization an all-inclusive tour for P8000

per person if not more than 100 persons join the tour. However, the

cost per person will be reduced by P50 for each person in excess of 100.

Find the largest possible gross sales revenue for the travel agency. How

many persons should join the tour and what is the cost per person so

that this maximum revenue is achieved?

please show how you got it, thank you for answering! it's greatly appreciated!

Answer 1

Let x be the number of persons who join the tour. Then, assuming x ≥ 100, the cost C per person is given by

C = 8000 - 50(x - 100)

C = 13000 - 50x

The revenue R is the per-person cost, times the number of persons:

R = xC = { 8000x if x ≤ 100

..............{13000x - 50x² if x ≥ 100

A graph of R readily shows that the maximum is achieved for some x > 100. So in this region,

dR/dx = 13000 - 100x

dR/dx = 0 ⇒ 13000 - 100x = 0 ⇒ x = 130

d²R/dx² = -100 < 0 so x = 130 gives a local max. But since R is a quadratic function of x, it has a single global maximum.

C(130) = 8000 - 50*(130 - 100)  = 6500P