Question

The daily profit in dollars of a specialty cake shop is described by the function P(x)=-6x^2 + 525x -1920 where x is the number of cakes prepared in one day. The maximum profit for the company occurs at the vertex of the parabola. How many cakes should be prepared per day in order to maximize profit?

a. 21 cakes
b. 42 cakes
c. 2646 cakes
d. 441 cakes

Answer 1

there are two ways to do this. either differentaite and find the maximum point. ie. P'(x)=-12x+525 at a stationary point this will equal zero. therefore 525=12x therefore x= 43.75 so ans is b. or you can complete the square and write the equation as p(x) = -6(x^2+525/-6)-1920
= -6((x+525/-12)^2-525^2/12^2)-1920
= -6(x+525/-12)^2-525^2/(12^2x-6)-1920
and then use transformation rules to find it is an y=X^2 graph then has been moved down by 1600.98... and stretched by a factor -6 in the y direction and the translated left by 43.75. therefore we can conclude that the x coordinate of the vertex is 43.75 since the vertex of the x^2 graph is zero. therefore x=43.75.

hope this helps

Answer 2

it DOES, thank you!