The owner of a luxury motor yacht that sails among the 4,000 Greek islands charges $600/person/day if exactly 20 people sign up for the cruise. However, if more than 20 people sign up (up to the maximum capacity of 90) for the cruise, then each fare is reduced by $5 for each additional passenger. Assume at least 20 people sign up for the cruise and let x denote the number of passengers above 20. Find a function R giving the revenue/day realized from the charter.

Question

The owner of a luxury motor yacht that sails among the 4,000 Greek islands charges $600/person/day if exactly 20 people sign up for the cruise. However, if more than 20 people sign up (up to the maximum capacity of 90) for the cruise, then each fare is reduced by $5 for each additional passenger. Assume at least 20 people sign up for the cruise and let x denote the number of passengers above 20. Find a function R giving the revenue/day realized from the charter. ​

Quizplus · Accepted Answer

Let's start by finding the fare for each passenger when more than 20 people sign up. For each additional passenger above 20, the fare is reduced by $5. So, if x passengers sign up above the initial 20, each of them will pay:

$600 - $5x

Therefore, the total revenue from x additional passengers will be:

(x)($600 - $5x) = $600x - $5x^2

Now, let's consider the initial 20 passengers. Since each of them pays $600, the total revenue from these passengers will be:

20($600) = $12,000

Therefore, the total revenue R as a function of x will be:

R(x) = $12,000 + ($600x - $5x^2)

Simplifying this expression, we get:

R(x) = -$5x^2 + $600x + $12,000

To find the maximum revenue, we need to find the vertex of the parabola given by this function. The x-coordinate of the vertex can be found using the formula:

x = -b/2a

where a = -5 and b = $600. Plugging in these values, we get:

x = -($600)/(2(-5)) = 60

So, the maximum revenue will be obtained when there are 60 passengers above the initial 20. To find the maximum revenue, we need to evaluate R(60):

R(60) = -$5(60)^2 + $600(60) + $12,000 = $42,000

Therefore, the maximum revenue is $42,000. Since this is the only possible answer choice that matches this value, the answer is (C).

The Owner of a Luxury Motor Yacht That Sails Among