PRECAL word problem Pizza Cost & Revenue

Question

Pizza Cost & Revenue

 Your school has contracted your favorite pizza parlor, Peppy’s Pizzeria, to cater lunch this Thursday.

You estimate initial costs at $80 to hire helpers, a delivery van, and the usual gas oven expenses. The cost for flour and toppings per each pizza is $2.00.

 1. Write a cost function that inputs n number of pizzas made and outputs your total catering costs. Costs = f(# of pizzas) = f(n).

2. How much will it cost you to make 20 pizzas for lunch on Thursday? Show this using your function.

 3. How many pizzas can you make on Thursday for $200?

 4. Make a graph of cost per pizza plus initial costs.

5. If you divide up each pizza into 8 slices and sell each slice for $1.00, write a function R that outputs the revenue received from selling n number of pizzas.

6. How much will you receive if you sell all 20 pizzas? Show this using your function.

7. On the same graph with your cost function, draw a graph of revenue per pizza sold.

8. Write a function P that outputs the profit from selling n pizzas.

9. How many pizzas did you need to sell to break even?

10.How many pizzas did you need to sell to make $100?

Answer 1

1. f(n)=2n+80, because if n=0 (no pizzas) the other charges would still apply. For every pizza the cost increases by $2, so the cost increases by 2n dollars if n pizzas are made.
2. When n=20, f(20)=40+80=$120. 
3. If f(n)=200, 200=2n+80, 2n=200-80=120, so n=60. The cost of 60 pizzas is $200.
4. Graph showing  red line:
5. R(n)=8n. When n=1, a single pizza is divided into 8 slices each costing $1. So n pizzas sell for 8n dollars.
6. When n=20, R(20)=$160, so you would receive $160 if you sold 160 slices at 8 $1 slices per pizza, 20 whole pizzas.
7. See blue line on the graph in question 4.
8. Profit P(n)=R(n)-f(n)=8n-2n-80=6n-80.
9. To break even, P(n)=0=6n-80, 6n=80, n=80/6=40/3=13⅓. If n=13, P(n)=-2, which is a loss of $2; if n=14, P(n)=$4, a small profit. If 14 pizzas were made, that would produce 112 slices, sold for $112. The cost of making 14 pizzas would be $108, so break even would be when 108 slices had been sold. Breakeven is where red and blue lines cross.
10. To make a profit of $100, P(n)=100=6n-80, 6n=180, n=30. So 30 pizzas would have to be made.