precalc word problem

Question

lyel's lemonade stand sells glasses of lemonade at baseball games for $1 a glass. at that price he sells 500 glasses of lemonade a game. hes noticed that for every 5 cents he raises the price, he sells 10 fewer glasses let x be the number of times lyle raises the price by 5 cents

a. what is the number of glasses sold in terms of a function of x? what is the price of each glass in terms of a function of x?

b. express the revenue as a function of x

c. what is the maximum revenue? at what price is the maximum revenue earned?

Answer 1

Hi Joann C.,

Were told to let x = # of times Lyle raises the price per glass by $.05.

Lets call P/G(x) = Price per Glass as a function of the number of price raises.

Let call G(x) = number of Glasses sold as a function of the number of price raises.

Let call R(x) = Revenue stream as a function of the number of price raises.

Then P/G(x) = $1.00 per glass plus the number of price raises times $.05, or:

P/G(x) = 1 + .05x, (this answers part a, price per glass)

And G(x) = 500 minus the number of price raises times 10, or:

G(x) = 500 - 10x, (this answers part a, number of glasses sold)

So R(x) = the number of glasses sold time the price per glass, or P/G(x)*G(x):

R(x) = P/G(x)*G(x) = (1 + .05x)*(500 - 10x)

R(x) = -.5x² + 15x + 500, (this answer part b).

R(x) is a parabola that opens down and its apex (-b/2a) is the maximum profit:

-b/2a = -15/[2*(-.5)] = 15

Therefore the maximum revenue is when x = 15, (I'll let you plug in this value and do the math).

R(15) = 612.5, (this is the maximum revenue for part c).

And the price per glass at maximum revenue is P/G(x), (again you can show the work):

P/G(15) = 1.75, (price per glass for part c).

I hope this helps, Joe.