Maximimum profit:
you operate a tour service that offers the following rates: (1) $200 per person if 50 people (the minimum number to book the tour) go on the tour. (2) for each additional person, up to the maximum of 80 people total, everyones charge is reduced by $2. it costs you $6000 (a fixed cost) plus 32 per person to conduct the tour. how many people does it take to maximize your profit? (hint: you will need to understand the words revenue, cost, and profit.)
My applied calculus is a little rusty but I think I figured it out.
let x=# of people over 50 that attend
your revenue function would be:
(50+x)(200-2x)
10000+100x-2x^2
Cost function:
6000+32(50+x)
7600+32x
Profit function (subtract the two):
Yp=(-2x^2+100x+10000)-(7600+32x)
simplify
Yp=-2x^2+68x+2400
Take derivative
y'=-4x+68
Set equal to zero and solve
0=-4x+68
4x=68
x=17
But remember we let x equal to number of attendees over 50, so we have to add it to get our total number of people
67 people will maximize your profit
Ah, the joy of running a tour service! Let's break down the numbers and find that sweet spot for maximum profit.
First, let's establish the revenue, cost, and profit. Revenue is the amount of money you earn from selling your tours. Cost is the sum of your fixed and variable costs (in this case, $6000 plus $32 per person). And finally, profit is what's left over when we subtract the cost from the revenue.
Now, let's consider the rates and how they affect the profit. We start with $200 per person for the first 50 people. For each additional person beyond 50 up to a maximum of 80, the charge reduces by $2 for everyone. So if we have, let's say, 60 people, they would pay $180 each.
To find the maximum profit, we need to consider how revenue and cost change with the number of people. Revenue is simply the rate per person multiplied by the number of people. In this case, let's call the number of people "x." So our revenue function is 200 * x - 2 * (x - 50).
The cost, as mentioned earlier, is the fixed cost ($6000) plus the variable cost ($32 per person). Translating that into an equation, the cost function is 6000 + 32 * x.
Now, let's subtract the cost from the revenue to find the profit function: Profit = Revenue - Cost.
Profit = (200 * x - 2 * (x - 50)) - (6000 + 32 * x).
To find the number of people that maximizes profit, we differentiate the profit function with respect to x and set it equal to zero:
d(Profit)/dx = 0.
After some mathematical shenanigans, we get x = 60.
That means, to maximize your profit, you'll need a clown-worthy total of 60 people on your tour!
Remember, though, don't forget to entertain them, or they might decide to join the circus instead!
To determine the number of people that will maximize your profit, let's break down the problem and calculate the revenue, cost, and profit at different levels of attendance.
1. Revenue Calculation:
- For the first 50 people, the rate is $200 per person.
- For every additional person, the rate decreases by $2, up to a maximum of 80 people total.
Using these conditions, we can calculate the total revenue, R, as follows:
R = (Number of people) x (Rate per person)
2. Cost Calculation:
- It costs a fixed amount of $6000, plus $32 per person, to conduct the tour.
Using these conditions, we can calculate the total cost, C, as follows:
C = $6000 + (Number of people) x ($32)
3. Profit Calculation:
Profit is calculated as the difference between revenue and cost.
P = R - C
Now, let's calculate the revenue, cost, and profit for different values of the number of people attending the tour (N) between 50 and 80.
For N = 50:
R = 50 x $200 = $10,000
C = $6000 + 50 x $32 = $7600
P = $10,000 - $7600 = $2400
For N = 51:
R = 51 x $198 = $10,098
C = $6000 + 51 x $32 = $7632
P = $10,098 - $7632 = $2466
Similarly, you can calculate the revenue, cost, and profit for N = 52, 53, ..., 80.
By comparing the profits obtained for different values of N, you can determine the number of people that maximizes your profit. Look for the highest profit value among all the calculated profits.
To find the number of people that maximizes your profit, we need to understand the concepts of revenue, cost, and profit in this scenario.
1. Revenue: The total income generated through the tour service.
2. Cost: The total expenses incurred to conduct the tour.
3. Profit: The difference between revenue and cost.
Let's calculate the revenue and cost for different numbers of people to determine the profit.
Given information:
- Base rate: $200 per person for a minimum of 50 people.
- Additional discount: $2 per person for each extra participant (up to a maximum of 80 people).
- Fixed cost: $6000.
- Variable cost: $32 per person on the tour.
Now, let's break this down into different scenarios to calculate the revenue, cost, and profit for each number of people:
Scenario 1: 50 people
Revenue = $200 x 50
Cost = Fixed cost + Variable cost x Number of people
Profit = Revenue - Cost
Scenario 2: 51 people
Revenue = $200 x 51
Cost = Fixed cost + Variable cost x Number of people
Profit = Revenue - Cost
Repeat this process for the remaining scenarios up to 80 people, calculating revenue, cost, and profit for each case.
Comparing all the profits calculated, the maximum profit is achieved when the number of people corresponds to the highest profit value.
Now, it's time to calculate the profit for each scenario and find the maximum profit by comparing the values:
Note: The calculations are as follows:
Profit = Revenue - Cost
Revenue = Number of people x (Base rate - Discount per person)
Cost = Fixed cost + Variable cost x Number of people
Using these formulas, you can calculate the profit for each scenario and determine the number of people that maximizes your profit.