Rational Functions Task

Question

Each summer, Coach Drummond runs a lacrosse camp for players from around the state. The amount that she charges per player is just enough to pay the expenses. The fixed cost of running the summer lacrosse camp is $14,200.

a)Complete the table with the amount that Coach Drummond would need to charge per player to cover the cost for different numbers of players at the camp.

Coach Drummond’s Lacrosse Camp

# of players/ amount charges per player

1

10

25

100

450

n

b) Last summer, in addition to the $14,200 fixed cost, the camp’s costs also included an extra $20 per player for meals and snacks. Write two equivalent expressions that represent the amount charged per player last summer for n players at the camp.

b)This summer, the camp’s costs are the same as last summer; however, Coach Drummond decides to offer 30 scholarships for players from around the state. Those 30 players will attend the camp for free, and all their costs will be paid for by the amount charged the other players. Create a function, c(n), that models the cost per player if n players attend the camp.

d)This summer, 450 players, including 30 who received scholarships, will attend Coach Drummond’s summer lacrosse camp. Next summer, the fixed cost is projected to increase to $15,700, and the cost for meals and snacks is projected to increase to $25 per player. Coach Drummond would like to keep the price per player the same as this summer. She anticipates that at least 450 players will attend next summer.

Write a recommendation to Coach Drummond that analyzes the number of players she will need to attend her camp next summer, the price per player, and the number of scholarships that she should offer. Provide evidence to support your answer.

Answer 1

What a problem! Let's take it step-by-step:

a) Complete the table with the amount that Coach Drummond would need to charge per player to cover the cost for different numbers of players at the camp.

The fixed cost of running the camp is $14,200, and the cost is completely covered by how much she charges each player. So if she only gets 1 player to join, their going to have to pay the entire $14,200 expense. Not likely. But if she can get 10 players to join, how much would each have to pay? That would be $14,200 divided up among 10 players, so $1,420 dollars. In general, then, the cost for n number of players to attend is simply 14200÷n. With that knowledge, we can fill the table.

______________

| 1| 14200.00|

| 10| 1420.00|

| 25| 568.00|

| 100| 142.00|

| 450| 31.56|

| n| 14200÷n|

|____|_________|

b) Write two equivalent expressions that represent the amount charged per player last summer for n players at the camp.

The cost now has an additional $20 per player. If n is the number of players, that means the additional cost is $20n. So the total cost is the fixed $14,200, plus the additional $20n (14200 + 20n). It still remains that each player has to pay their fair share however, so the cost for each player will be that new total, divided by n again. Then, we can rewrite that expression into a more comfortable form as follows:

14200 + 20n 14200 20n 14200

————————————— = ——————— + ————— = ——————— + 20

n n n n

c) Create a function, c(n), that models the cost per player if n players attend the camp.

Now 30 players get to attend for free, and the cost is shouldered by the rest of the n-30 players. So to modify our nice cost per player expression we derived above, all we need is to, instead of all n players, divide by n-30 players:

14200

c(n) = ———————— + 20

n - 30

d) Write a recommendation to Coach Drummond that analyzes the number of players she will need to attend her camp next summer, the price per player, and the number of scholarships that she should offer. Provide evidence to support your answer.

This is the trickiest part. First of all, Coach Drummond wants the player's price for next year to be the same as it was for this year, so find out how much she charged her 450 players this year, by plugging 450 into our cost equation. We find that c(450) is $53.81, rounded to the nearest cent. So this year, when the total cost is increased to $15,700, plus $25 per player, how many players would she need to keep the price the same? We can find that by modifying function c(n) to c´(n), and solving for n:

15700 15700 15700

c´(n) = ——————— + 25 => 53.81 = ——————— + 25 => n = ——————— ≈ 545

n n 28.81

So Coach Drummond would need 545 students in order to keep the price the same. Here's where the subjective part comes in. She is only pretty sure that 450 players will attend the following year, but if she sets the price at $53.81, and she doesn't get 545 players, she loses money. So perhaps she should give out a certain number of scholarships to increase enrollment, but raise the price a bit to make up for the scholarships.

Ultimately, you'll have to write your own recommendation, but hopefully this has helped you think about the problem. And here are a few more things to consider to help in writing your recommendation:

1. If Coach Drummond is sure she'll have 450 players, who don't mind the cost increase, what's the smallest number of scholarships she can give out to reach 545 player enrollment?
2. If she gives out this number of scholarships, how does that change the cost function c´(n)? (Hint: it's very similar to the c(n) cost function!)
3. Finally, given this new cost function, what is the new cost for each player?

Good luck! :)