The owner of a luxury motor yacht that sails among the 4000 Greek islands charges $630 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 $5 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) =
(b) What is the revenue per day if 42 people sign up for the cruise?
$
(c) What is the revenue per day if 65 people sign up for the cruise?
$
R(x) = 630*20 + x(630-5x) for 0 < x <= 70
R(42) = 630*20 + 22*(630-5*22) = 24040
R(65) = 630*20 + 45*(630-5*45) = 30825
To find the revenue per day realized from the charter, we need to consider the number of passengers above 20 (denoted by x) and the fare per person per day.
Since the fare per person per day is initially $630, if there are exactly 20 people on board, the revenue per day would be 20 * $630 = $12,600.
Now, for each additional passenger (x), the fare per person per day is reduced by $5. So, the fare per person per day for the additional passengers would be $630 - $5x.
To find the total revenue per day, we multiply the number of passengers by their respective fares.
Therefore, the function R(x) for the revenue per day is:
R(x) = (20 + x) * (630 - 5x)
To find the revenue per day if 42 people sign up for the cruise, we substitute x = 42 - 20 = 22 into the function R(x):
R(22) = (20 + 22) * (630 - 5*22)
Simplifying this expression will give us the revenue per day when 42 people sign up for the cruise.
To find the revenue per day if 65 people sign up for the cruise, we substitute x = 65 - 20 = 45 into the function R(x):
R(45) = (20 + 45) * (630 - 5*45)
Again, simplifying this expression will give us the revenue per day when 65 people sign up for the cruise.
(a) To find the function R giving the revenue per day realized from the charter, we need to consider two scenarios: when x is less than or equal to 70 (90 - 20) and when x is greater than 70.
For x less than or equal to 70, each fare is reduced by $5 for each additional passenger. We can express this as (20 + x) fares being reduced by $5x in total. The fare per person per day is $630, so the revenue per day in this scenario is:
R(x) = (20 + x) * ($630 - $5x)
For x greater than 70, the fare per person per day will be at the maximum reduction, which is $30 less than the initial $630 fare. Therefore, the revenue per day in this scenario is:
R(x) = (20 + 70) * ($630 - $5 * 70) - $30 * (x - 70)
So the overall function R giving the revenue per day realized from the charter becomes:
R(x) = (20 + x) * ($630 - $5x) for x ≤ 70,
R(x) = (20 + 70) * ($630 - $5 * 70) - $30 * (x - 70) for x > 70.
(b) To find the revenue per day if 42 people sign up for the cruise, we substitute x = 42 into the function R:
R(42) = (20 + 42) * ($630 - $5 * 42)
Simplifying the expression:
R(42) = 62 * ($630 - $210)
R(42) = 62 * $420
The revenue per day when 42 people sign up for the cruise is $26,040.
(c) To find the revenue per day if 65 people sign up for the cruise, we substitute x = 65 into the function R:
R(65) = (20 + 65) * ($630 - $5 * 65) for x ≤ 70
R(65) = (20 + 70) * ($630 - $5 * 70) - $30 * (65 - 70) for x > 70
Simplifying the expression:
R(65) = 85 * ($630 - $325)
R(65) = 85 * $305
The revenue per day when 65 people sign up for the cruise is $25,925.