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?
$ ?
To find the function R(x) giving the revenue per day realized from the charter, we need to consider the number of passengers above 20 (x).
First, let's determine the fare per person per day. For exactly 20 people signing up, the charge is $430 per person per day. However, if more than 20 people sign up (up to a maximum of 90), the fare is reduced by $4 per day for each additional passenger.
Therefore, if x passengers sign up above the initial 20, the fare per person per day will be $430 - ($4 * x).
Now, we can calculate the total revenue per day by multiplying the fare per person per day by the total number of passengers (20 + x):
R(x) = (20 + x) * ($430 - $4x)
Simplifying, we get:
R(x) = -4x^2 + 350x + 8600
(a) To find the revenue per day if 47 people sign up for the cruise, substitute x = 47 into the function R(x):
R(47) = -4(47)^2 + 350(47) + 8600
R(47) ≈ $19,220
Therefore, the revenue per day if 47 people sign up for the cruise is approximately $19,220.
(b) To find the revenue per day if 84 people sign up for the cruise, substitute x = 84 into the function R(x):
R(84) = -4(84)^2 + 350(84) + 8600
R(84) ≈ $25,840
Therefore, the revenue per day if 84 people sign up for the cruise is approximately $25,840.
(a) To find the function R that gives the revenue per day realized from the charter, we need to consider the different scenarios based on the number of passengers.
If there are exactly 20 people signed up, the fare per person per day is $430. In this case, the revenue per day would be:
R(0) = 20 * $430 = $8600
For each additional passenger above 20, the fare per person per day is reduced by $4. So, if x represents the number of passengers above 20, the revenue per day can be represented by the equation:
R(x) = (20 + x) * ($430 - $4x)
Simplifying this equation, we get:
R(x) = -4x^2 + 350x + 8600
Therefore, the function R that gives the revenue per day realized from the charter is R(x) = -4x^2 + 350x + 8600.
(b) If 47 people sign up for the cruise, we need to find the value of R(x) when x = 47 - 20 = 27.
Substituting x = 27 into the equation R(x), we get:
R(27) = -4(27)^2 + 350(27) + 8600
= -4(729) + 9450 + 8600
= -2916 + 9450 + 8600
= 15134
The revenue per day when 47 people sign up for the cruise is $15,134.
(c) If 84 people sign up for the cruise, we need to find the value of R(x) when x = 84 - 20 = 64.
Substituting x = 64 into the equation R(x), we get:
R(64) = -4(64)^2 + 350(64) + 8600
= -4(4096) + 22400 + 8600
= -16384 + 22400 + 8600
= 74616
The revenue per day when 84 people sign up for the cruise is $74,616.