Question

the owner of a luxury motor yacht that sails among the 4000 greek islands charges $568/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 100) for the cruise, then each fare is reduced by $4 for each additional passenger. assuming at least 20 people sign up for the cruise, determine how many passengers will result in the maximum revenue for the owner of the yacht.

Answer 1

Answer:26.72

Step-by-step explanation: To determine the number of passengers that will result in the maximum revenue for the owner of the yacht, we can set up an equation and find the derivative to find the maximum point.

Let's assume the number of additional passengers above 20 is represented by 'x'. We can calculate the fare per person using the given information:

Fare per person = $568 - $4x

The total revenue can be calculated by multiplying the fare per person by the total number of passengers:

Revenue = (20 + x) * (568 - 4x)

To find the maximum revenue, we can take the derivative of the revenue function with respect to 'x' and set it equal to zero:

d(Revenue)/dx = 0

Differentiating the revenue function:

d(Revenue)/dx = -8x^2 + 168x + 11360

Setting the derivative equal to zero:

-8x^2 + 168x + 11360 = 0

Now we can solve this quadratic equation to find the value of 'x' that results in maximum revenue.

Using the quadratic formula:

x = (-b ± √(b^2 - 4ac))/(2a)

For our equation, a = -8, b = 168, and c = 11360.

x = (-168 ± √(168^2 - 4(-8)(11360)))/(2(-8))

Simplifying this equation will give us the values of 'x' that will result in maximum revenue.

After calculating, we find that x ≈ 26.72 or x ≈ -4.22.

Since the number of passengers cannot be negative, we take x ≈ 26.72.