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. correct

b. put x=27, solve

c. put x=64, solve

Answer 2

(a) To find the function R giving the revenue per day realized from the charter, we need to consider the different scenarios based on the number of passengers.

If "x" represents the number of passengers above 20, then the total number of passengers would be 20 + x.

For each additional passenger above 20, the fare is reduced by $4 per day. So the revenue per day for each additional passenger is $430 - $4x.

To find the total revenue per day, we multiply the number of passengers by the fare per person per day:

R(x) = (20 + x) * ($430 - $4x)

Expanding and simplifying the expression:

R(x) = 20 * $430 + 20 * (-$4x) + x * ($430 - $4x)
= $8600 - $80x + $430x - $4x^2
= - $4x^2 + $350x + $8600

Therefore, the function R giving the revenue per day realized from the charter is R(x) = -4x^2 + 350x + 8600.

(b) To find the revenue per day if 47 people sign up for the cruise, we substitute x = 47 - 20 = 27 into the function R(x):

R(27) = -4(27)^2 + 350(27) + 8600
= -4(729) + 9450 + 8600
= -2916 + 9450 + 8600
= 5134

So, the revenue per day if 47 people sign up for the cruise is $5134.

(c) To find the revenue per day if 84 people sign up for the cruise, we substitute x = 84 - 20 = 64 into the function R(x):

R(64) = -4(64)^2 + 350(64) + 8600
= -4(4096) + 22400 + 8600
= -16384 + 22400 + 8600
= 7456

So, the revenue per day if 84 people sign up for the cruise is $7456.