john owns a hot dog stand. his profit in dollars is given by the equation P(x)=P=-x^2+14x+54, where x is the number of hot dogs sold. what is the most he can earn?

## find profit

# 2 Answers

okay, that's an upside-down U and the maximum value is at the top

if you put the function in vertex form you can easily find the vertex:

group the first two terms and factor out the negative (essentially factoring out -1):

p = - (x²-14x) + 54

complete the square by taking the middle coefficient (14), dividing it by 2, squaring it and adding the result to the terms in parentheses

p = -(x²-14x+49) + 54 + 49 (we've really subtracted 49, so we have to undo this by adding 49)

p = -(x-7)² + 103

the vertex is at (7,103), so your maximum profit is **$103**

# Comments

thank you...thats what i got ...but i have been having trouble so i was worried!! thanks!

P(x)=P=x^{2}+14x+54

Function y=ax^{2}+bx+c is a parabola with vertex at x= -b/2a. It opens up if a>0 and it opens down if a <0.

So function P(x)=P=x^{2}+14x+54 is a parabola with a=1, b=14, c=54. Since a=1 >0, it opens up.

Its vertex is at

x = -b/2a = -14/2 = -7

P(-7) = (-7)^{2}+14(-7)+54 = 49-98+54 = $5

So the vertex is at (-7, 5)

P(-7) = $5 is the lowest profit point of the parabola since parabola opens up. From this point profit increases in either direction. But negative x values have no meaning in this application and when no hot dogs are sold (x=0), John makes a profit of P(0)=0^{2}+14(0)+54=$54
!!!!

Since the parabola is opening up, you can see that as x (number of hotdogs) goes up, profit increases and there is no limit to the profit.

**Second way to solve this problem is using derivatives.**

This function will have maxima (highest profit) or minima (lowest profit) when its derivative = 0

dP/dx = 2x + 14 = 0

2x + 14 = 0

- 14 -14

2x = -14

x = -7

P(-7) = (-7)^{2}+14(-7)+54 = 49-98+54 = $5

So the vertex is at (-7, 5)

The parabola opens up, if its second derivative is positive otherwise it opens down.

d^{2}P/dx = d/dx (2x+14) = 2

Since second derivative is positive, parabola opens up so the first derivative gives us the minimum for profit of $5 and there is no limit to profit since profit increases as number of hot dogs sold increases.

## Comments

is that really positive x²? your profit function is a quadratic function which opens up..that means it goes up to infinity.

sorry! you're correct ..its negative

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