john owns a hotdog stand. his profit is represented by the equation P(x)=-x^2+14x+58, where x is the number of hotdogs. What is the most he can earn?

Question

I cannot figure out the steps to this problem.

Answer 1

To put it simply, when you see a quadratic equation like this, always try to re-write the expression as a square form involving x. Since anything squared can't be negative, its minimum value is zero.

 

Here, P(x) = -x²+14x+58

 

                  =-(x²-2.7.x+7²)+107

 

                  =-(x-7)²+ 107

 

                   = 107 - (x-7)²

 

Value of this expression is maximum when (x-7)²  is minimum because that is what we are subtracting from 107.

As we discussed before, anything squared can have a minimum value of zero, which is, when x = 7 or he sells 7 hot dogs. So, in other words, when John sells 7 hotdogs, his earnings are maximum. That maximum earning is $107.

Answer 2

The most he can earn is the maximum P(x).  We can find this out by finding the vertex of P(x).  Any function in vertex form is

 

P(x) = a(x - h)² + k

 

where:

a is the coefficient of x² term

The vertex is the coordinate (h, k)

 

P(x) = -(x² - 14x + 49) + 107

P(x) = -(x - 7)(x - 7) + 107

P(x) = -(x - 7)² + 107

 

The vertex is (7, 107).  From this vertex, the most John can earn is $107.