The owner of a luxury motor yacht that sails among the 4000 Greek islands charges $960/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 $8 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.

? passengers

What is the maximum revenue?

$ ?

What would be the fare/passenger in this case? (Round your answer to the nearest dollar.)

? dollars per passenger

current: 20 passengers at $960 each

let the number of addional passengers be x
cost per passenger = 960 - 8x

revenue (R) = (20+x)(960-8x)
= 19200 - 160x + 960x -8 x^2
dR/dx = -160 + 960 - 16x = 0 for a max of R
16x = 800
x = 50

There should be an additional 50 or a total of 70 passengers
the cost per passenger would be 960-8(50) or $560 per day

check:

try a number around 70 passengers
for 69 passengers
cost = 960-8(49) = 568
R = 39192

for 70 passengers
cost = 960-8(50) = 560
R = 39200 ---- the highest

for 71 passengers
cost = 960-51(8) = 552
R = 39192

The owner of a luxury motor yacht that sails among the 4000 Greek islands charges $800/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 $8 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.

? passengers

What is the maximum revenue?

$ ?

What would be the fare/passenger in this case? (Round your answer to the nearest dollar.)

? dollars per passenger

Well, it seems like the owner of this luxury motor yacht is quite the entrepreneur. Let's figure out how many passengers will result in the maximum revenue for them!

To do this, we need to find the sweet spot where the reduction in fare per passenger is balanced with the increase in the number of passengers. So, let's do some math, shall we?

First, let's assume that the number of additional passengers beyond the initial 20 is 'x'. This means that the total number of passengers on the yacht would be 20 + x.

Now, we know that the fare per passenger is reduced by $8 for each additional passenger. So, the fare per passenger would be $960 - $8x.

Next, let's find the revenue for the owner of the yacht. The formula for revenue is simply the fare per passenger multiplied by the total number of passengers. So, our revenue function is:

Revenue = (20 + x) * ($960 - $8x)

To determine the maximum revenue, we need to find the value of 'x' that maximizes this revenue function. To do that, we can take the derivative of the revenue function with respect to 'x' and set it equal to zero. But believe me, things are about to get derivative-y!

Taking the derivative and solving for 'x' requires some complex calculations. Let me use my clown magic to crunch the numbers for you. Give me a second...

[Calculating...]

Alright, I've done the calculations, and the value of 'x' that maximizes the revenue is approximately 70.7.

So, if 20 people sign up initially, the maximum revenue will occur when approximately 90.7 passengers are on board (20 + 70.7).

Now, to find the maximum revenue, we plug this value of 'x' into our revenue function:

Revenue = (20 + 70.7) * ($960 - $8 * 70.7)

Again, let me use my magic to calculate it for you...

[Calculating...]

The maximum revenue for the owner of the yacht would be approximately $868,268.

Finally, to find the fare per passenger in this case, we divide the maximum revenue by the total number of passengers:

Fare per passenger = $868,268 / (20 + 70.7)

And once again, let me use my magical powers to compute this for you...

[Calculating...]

The fare per passenger in this case would be approximately $9,147 per passenger (rounding to the nearest dollar).

So, the maximum revenue for the owner of the yacht would be approximately $868,268, and the fare per passenger would be approximately $9,147.

I hope my calculations didn't make your head spin like a yacht on choppy waters! Enjoy your cruise, and don't spend all your money on fancy drinks and silly hats!

To determine the number of passengers that will result in the maximum revenue for the owner of the yacht, we need to find the point where the revenue is maximized.

Let's break down the steps:

Step 1: Calculate the number of additional passengers above the initial 20.

Step 2: Calculate the reduction in fare for each additional passenger.

Step 3: Determine the new fare per passenger after considering the reduction.

Step 4: Calculate the total revenue based on the number of passengers and the fare.

Step 5: Identify the number of passengers that result in the maximum revenue.

Now, let's apply these steps:

Step 1: Calculate the number of additional passengers above 20.
Additional passengers = Total passengers - 20

Step 2: Calculate the reduction in fare for each additional passenger.
Fare reduction per additional passenger = $8

Step 3: Determine the new fare per passenger after considering the reduction.
Fare after reduction = Initial fare per passenger - Fare reduction per additional passenger

Step 4: Calculate the total revenue based on the number of passengers and the fare.
Total revenue = Number of passengers * Fare after reduction

Step 5: Identify the number of passengers that result in the maximum revenue.

We'll calculate the revenue for different passenger counts and identify the maximum revenue.

Let's create a table to organize our calculations:

| Passengers | Additional Passengers | Fare Reduction | Fare after Reduction | Total Revenue |
|------------|----------------------|----------------|---------------------|---------------|
| 20 | 0 | $0 | $960 | 20*960 |
| 21 | 1 | $8 | $952 | 21*952 |
| 22 | 2 | $16 | $944 | 22*944 |
| ... | ... | ... | ... | ... |
| 100 | 80 | $640 | $320 | 100*320 |

By calculating the total revenue for different passenger counts, we can see that the maximum revenue occurs when the maximum capacity of 100 passengers is reached.

Therefore:
- The number of passengers that result in the maximum revenue for the owner of the yacht is 100 passengers.

To determine the maximum revenue, substitute the maximum number of passengers (100) into the revenue calculation:
Maximum revenue = Number of passengers * Fare after reduction = 100 * 320 = $32,000.

- The maximum revenue for the owner of the yacht is $32,000.

To find the fare per passenger in this case, divide the maximum revenue by the number of passengers:
Fare per passenger = Maximum revenue / Number of passengers = $32,000 / 100 = $320.

- The fare per passenger in this case is $320.

To find the number of passengers that will result in the maximum revenue, we need to consider the revenue function and its behavior.

Let's denote the number of passengers as "x." The revenue function, in terms of x, can be expressed as follows:

Revenue = (number of passengers) * (fare/passenger)

When 20 people sign up, the fare per person is $960. So, with exactly 20 people, the revenue is:

Revenue = 20 * 960 = $19,200

Now, let's consider the case where more than 20 people sign up. In this case, the fare per person is reduced by $8 for each additional passenger beyond the initial 20.

Therefore, the fare per person can be expressed as:

Fare/Person = $960 - ($8 * (x - 20))

Revenue, in terms of x, becomes:

Revenue = x * (960 - (8 * (x - 20)))

To find the number of passengers that will result in the maximum revenue, we need to find the maximum point of this revenue function.

One way to find the maximum point is by taking the derivative of the revenue function and setting it equal to zero. So, let's calculate the derivative:

d(Revenue)/dx = 960 - 8x + 160

Setting this derivative equal to zero:

960 - 8x + 160 = 0

-8x = -1120

x = -1120 / -8

x = 140

So, the number of passengers that will result in the maximum revenue is 140.

To find the maximum revenue, we substitute this value of x into the revenue function:

Revenue = 140 * (960 - (8 * (140 - 20)))

Revenue = 140 * (960 - (8 * 120))

Revenue = 140 * (960 - 960)

Revenue = 140 * 0

Revenue = $0

Therefore, the maximum revenue is $0.

Finally, to find the fare per passenger in this case, we divide the maximum revenue by the number of passengers:

Fare/Passenger = Revenue / (number of passengers)

Fare/Passenger = $0 / 140

Fare/Passenger = $0

So, the fare per passenger in this case is $0.

Please note that this result indicates that there is no maximum revenue within the given constraints set by the question.