Charter Revenue The owner of a luxury motor yacht that sails among the 4,000 Greek islands charges $\$ 610$ 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 $\$ 7$ 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 (in dollars) realized from the charter.
\[
R(x)=
\]
(b) What is the revenue per day (in dollars) if 49 people sign up for the cruise?
\[
\$ \square x
\]
(c) What is the revenue per day (in dollars) if 84 people sign up for the cruise?
\[
\$
\]
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Answer
Therefore, the revenue per day when 84 people sign up for the cruise is \( \boxed{37128} \).
Step 1 ：The revenue per day can be calculated using the function:
Step 2 ：\( R(x) = \begin{cases} (610 - 7x)(20 + x) & \text{if } 20 < x \leq 90 \\ 12200 & \text{if } x \leq 20 \end{cases} \)
Step 3 ：To find the revenue per day when 49 people sign up for the cruise, substitute \( x = 49 - 20 = 29 \) into the function:
Step 4 ：\( R(29) = (610 - 7(29))(20 + 29) = 581 \cdot 49 = 28469 \)
Step 5 ：Therefore, the revenue per day when 49 people sign up for the cruise is \( \boxed{28469} \).
Step 6 ：To find the revenue per day when 84 people sign up for the cruise, substitute \( x = 84 - 20 = 64 \) into the function:
Step 7 ：\( R(64) = (610 - 7(64))(20 + 64) = 442 \cdot 84 = 37128 \)
Step 8 ：Therefore, the revenue per day when 84 people sign up for the cruise is \( \boxed{37128} \).
