SOLUTION: John owns a hotdog stand. He has found that his profit is represented by the equation {{{P(x)= -x^2+56x+73}}},, where x is the number of hotdogs. How many hotdogs must he sell to m

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: John owns a hotdog stand. He has found that his profit is represented by the equation {{{P(x)= -x^2+56x+73}}},, where x is the number of hotdogs. How many hotdogs must he sell to m      Log On


   

Question 114677: John owns a hotdog stand. He has found that his profit is represented by the equation P%28x%29=+-x%5E2%2B56x%2B73,, where x is the number of hotdogs. How many hotdogs must he sell to make a profit?
P(x)=x^2+56x+73
Choices:
45 hotdogs
28 hotdogs
56 hotdogs
73 hotdogs
Thank you.
Answer by edjones(8007) About Me  (Show Source):
You can put this solution on YOUR website!
The question is not phrased properly.
One formula has -x^2 in it and the repeat of it has x^2.
The question should be "how many hot dogs must he sell to maximize his profit".
P%28x%29=+-x%5E2%2B56x%2B73
x=-1.27456, x=57.27456 quadratic formula (see below)
(-1.27456+57.27456)/2=28 maximum of the graph
By checking the next graph you can see that he can make a profit by selling between 1 and 57 hot dogs and he makes a maximum profit at 28 hot dogs.
graph%28500%2C500%2C-10%2C75%2C-50%2C1000%2C-x%5E2%2B56x%2B73%29
Ed
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case -1x%5E2%2B56x%2B73+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%2856%29%5E2-4%2A-1%2A73=3428.

Discriminant d=3428 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-56%2B-sqrt%28+3428+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%2856%29%2Bsqrt%28+3428+%29%29%2F2%5C-1+=+-1.27456233660889
x%5B2%5D+=+%28-%2856%29-sqrt%28+3428+%29%29%2F2%5C-1+=+57.2745623366089

Quadratic expression -1x%5E2%2B56x%2B73 can be factored:
-1x%5E2%2B56x%2B73+=+%28x--1.27456233660889%29%2A%28x-57.2745623366089%29
Again, the answer is: -1.27456233660889, 57.2745623366089. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-1%2Ax%5E2%2B56%2Ax%2B73+%29