The owner of a luxury motor yacht that sails among the 4000 Greek islands charges $430 per person per day if exactly 20 people sign up for the cruise. However, if more than 20 people (up to the maximum capacity of 90) sign up for the cruise, then each fare is reduced by $4 per day for each additional passenger. Assume at least 20 people sign up for the cruise, and let x denote the number of passengers above 20.

Question

The owner of a luxury motor yacht that sails among the 4000 Greek islands charges $430 per person per day if exactly 20 people sign up for the cruise. However, if more than 20 people (up to the maximum capacity of 90) sign up for the cruise, then each fare is reduced by $4 per day for each additional passenger. Assume at least 20 people sign up for the cruise, and let x denote the number of passengers above 20.

(a) Find a function R giving the revenue per day realized from the charter.

R(x) = -4x^2+350x+8600

(b) What is the revenue per day if 47 people sign up for the cruise?

$ ??

(c) What is the revenue per day if 84 people sign up for the cruise?

$ ??

Answer 1

(a) To find the revenue function R(x), we need to determine how the revenue per day changes when the number of passengers above 20 (x) increases.

We know that the fare for each passenger is initially $430 per day when exactly 20 people sign up. For each additional passenger (x), the fare is reduced by $4 per day. So the fare per day for x passengers above 20 is given by:
Fare per day = $430 - $4x

To find the total revenue per day, we need to multiply the fare per day by the total number of passengers (20 + x):
R(x) = (20 + x) * (Fare per day)
R(x) = (20 + x) * ($430 - $4x)

Expanding and simplifying the expression:
R(x) = 20*$430 + 20*(-$4x) + x*$430 + x*(-$4x)
R(x) = $8600 - $80x + $430x - $4x^2
R(x) = -4x^2 + $350x + $8600

So, the revenue function R(x) is given by R(x) = -4x^2 + $350x + $8600.

(b) To find the revenue per day when 47 people sign up for the cruise, we substitute x = 47 into the revenue function R(x):
R(47) = -4(47)^2 + $350(47) + $8600
R(47) = -4(2209) + $16450 + $8600
R(47) = -$8836 + $16450 + $8600

Calculating the expression:
R(47) = $16560

Therefore, the revenue per day when 47 people sign up for the cruise is $16,560.

(c) To find the revenue per day when 84 people sign up for the cruise, we substitute x = 84 into the revenue function R(x):
R(84) = -4(84)^2 + $350(84) + $8600
R(84) = -4(7056) + $29400 + $8600
R(84) = -$28224 + $29400 + $8600

Calculating the expression:
R(84) = $45000

Therefore, the revenue per day when 84 people sign up for the cruise is $45,000.