The Toro Ski club is planning a ski trip for the upcoming long weekend. They have 40 skiers signed up to go, and the ski resort is charging $120 per person. The resort manager offers to lower the rate per person by $2 for each additional skier up to a maximum number m. (So if they get 41 people to go, the rate will be $118 per person. How many skiers should go for the resort to bring in the most revenue?
right now:
40 people signed up, $120 per person.
let the number of additional skiers be m
number going : 40+m
cost per skier : 120-2m
revenue = (40+m)(120-2m)
= 4800 + 40m - 2m^2
This is a quadratic, a parabola opens down, so it has a maximum revenue.
This is found by finding the vertex.
Do that by the means that were taught to you.
To determine the number of skiers that would generate the most revenue for the resort, we need to find the value of m, which represents the maximum number of additional skiers after the initial 40.
Let's break down the problem step by step:
1. Determine the rate per person: The rate per person is $120 initially, and for each additional skier, it decreases by $2. So, for x additional skiers, the rate per person can be calculated as:
Rate per person = $120 - ($2 * x)
2. Calculate the total revenue for the resort: The total revenue is the product of the rate per person and the number of skiers. So, the total revenue can be expressed as:
Revenue = (40 + x) * (Rate per person)
3. Find the value of m: As mentioned in the question, the rate per person decreases until a maximum number of additional skiers, m, is reached. We need to determine the value of m that maximizes the revenue.
To find the value of m, we can create an equation for the revenue and then find the maximum point by taking the derivative:
Revenue = (40 + x) * (120 - 2x)
Taking the derivative of the revenue function:
d(Revenue)/dx = 120 - 4x
To find the maximum revenue, set the derivative equal to zero and solve for x:
120 - 4x = 0
4x = 120
x = 30
Therefore, the maximum number of additional skiers is 30. To maximize revenue, the total number of skiers should be 40 + 30 = 70.
So, the Toro Ski club should aim to have 70 skiers in order to generate the most revenue for the resort.