The function f(x) = -x^2 + 24x-80 models the hourly profit, in dollars a shop makes for selling coffee, where x is the number of cups of coffee sold, and f(x) is the amount of profit. part a: determine the vertex. what does this calculation mean in the context of this problem? part b: deterrmine the x-intercepts. what do these values mean in the context of this problem?

Remark

The first think you must do is complete the square. That will give you the vertex. What that means in terms of this problem will be discussed when we have the vertex.

Step One

Find the vertex.

y = -x^2 + 24x - 80

y = (-x^2 + 24x) - 80

y = -1 (x^2 - 24x) - 80

y = -1 (x^2 - 24x + (24/2)^2 ) - 80 + (24/2)^2 Follow what the minus sign on the left side of the brackets does inside the brackets.

y = -1(x^2 - 24x + 144) - 80 + 144

y = -1(x -12)^2 + 64

The vertex of the original equation is (12,64)

Step two

Answer what the vertex means.

It means that if 12 cups of coffee are sold in an hour, the maximum profit is 64 dollars. Any other number will kick -1(x - 12)^2 into play and subtract from 64.

Step three

What are the x intercepts.

make y = 0

- (x - 12)^2 + 64 = 0

-(x - 12)^2 = - 64 Divide by -1

(x - 12)^2 = 64 Take the square root of both sides.

x - 12 = +/-8 Add 12 to both sides

x = 12 +/- 8

x = 12 + 8

x = 20 or

x = 12 - 8

x = 4

Step Four

Interpret the results.

If he sells as many as 20 cups of coffee he makes no profit.

If he sells as few as 4 cuts of coffee he also makes no money.

It's an odd situation but that's what you get.

25% is the answer