Question

John owns a hotdog stand. His profit is represented by P(x)=-x^2+10x+34,with p (x) being profit and X the number of hotdogs sold. What is the most he can earn in dollars?

Answer 1

Answer: $57

Explanation:

Answer 2

$59

EXPLANATION

John's profit is represented by a quadratic function, whose leading coefficient is negative. This means that the graph is a parabola that opens downward and, therefore, the vertex is a maximum. In other words, the most profit he can earn is the y-coordinate of the vertex.

The x-coordinate of the vertex of a quadratic function given in standard form is,

`\begin{gathered} f(x)=ax^2+bx+c \\ x_(vertex)=(-b)/(2a) \end{gathered}`

In this case, a = -1 and b = 10,

`x_(vertex)=(-10)/(2(-1))=(-10)/(-2)=5`

And the maximum profit is given by P(x_vertex),

`P(x_(vertex))=P(5)=-5^2+10\cdot5+34=-25+50+34=59`

Hence, the most he can earn is $59.