SOLUTION: Lucy’s Perfect Pizza sells every pizza for $12. Lucy currently has 400 customers per day. She is considering raising the price for each pizza in order to maximize her daily incom

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Question 1184238: Lucy’s Perfect Pizza sells every pizza for $12. Lucy currently has 400 customers per day. She is considering raising the price for each pizza in order to maximize her daily income. She estimates that the business will lose 10 customers per day for every $0.50 in the price of the pizza. Write a function to represent Lucy’s daily income as a function of the number of $0.50 increases in the price of pizza. Determine the x- and y-intercepts and explain what each represents in the context of the problem situation. Determine the maximum daily income for Lucy’s Perfect Pizza, the corresponding pizza price, and the corresponding number of daily customers. Thank you!
Found 2 solutions by Theo, ikleyn:
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
y = (12 + .5x) * (100 - 10x)
graph looks like this.

from the graph, you can see.....

when x = 0, the revenue is 12 * 400 = 4800.

when x = 8, the revenue is (12 + 4 = 16) * (400 - 80 = 320) = 5120.

that's the maximum revenue, based on the equation.

when x = 20, the revenue is (12 + 10 = 22) * (400 - 200 = 200) = 4400.

the y-intercept is the value of y when x = 0.

that's 12 * 400 = 4800 income.

the x-intercept is the value of x when y = 0.

that occurs when x = 40.

the price is 12 + 20 = 31.
the number of customers is 400 - 400 = 0.
the revenue is 31 * 0 = 0.
when x = 40, there are no customers left to buy a pizza.

the value of x and the revenue are shown on the graph.
x = the number of increments of .5 added to the price and the number of increments of 10 subtracted from the quantity.
y is the total revnue.
if you want to know what the price is at these points, you have to determine it from the eqution.
when x = 8, price = 12 + .5*8 = 12 + 4 = 16
when x = 8, quantity = 400 - 10*8 = 400 - 80 = 320.
this evaluation of the equation is done for each increment of x that you want to see.
the graph does give you a visual of the shape of the revenue runction and when the maximum revenue is attained.

Answer by ikleyn(52851) (Show Source):
You can put this solution on YOUR website!
.
Lucy’s Perfect Pizza sells every pizza for $12. Lucy currently has 400 customers per day.
She is considering raising the price for each pizza in order to maximize her daily income.
She estimates that the business will lose 10 customers per day for every $0.50 in the price of the pizza.
Write a function to represent Lucy’s daily income as a function of the number of $0.50 increases in the price of pizza.
Determine the x- and y-intercepts and explain what each represents in the context of the problem situation.
Determine the maximum daily income for Lucy’s Perfect Pizza, the corresponding pizza price,
and the corresponding number of daily customers. Thank you!
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The price of the pizza after n increases of the price p = 12 + 0.5*n dollars. The number of the customers after n increases of the price N = 400 - 10n. The income (the revenue function) R = p*N = (12 + 0.5n)*(400-10n). X-intercepts are n = = 40 and n = = -24. Optimal number of increases is exactly half-way between the x-intercepts = = = 8. Optimal number of customers = 400 - 10*8 = 400 - 80 = 320. Optimal price = 12 + 0.5*8 = 16. Maximum daily income = = 320*16 = 5120 dollars.

Solved.