MHB Revenue and Cost and Profit Problem

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Given the following revenue and cost functions, find the maximum profit.

R(x) = 71x - 2x^2; C(x) = 23x + 108

P(x) = Revenue - Cost

P(x) = 71x-2x^2 - (23x+108)

P(x) = -2x^2 + 48x - 108 *I am stuck right here. Using quadratic formula but cannot seem to solve it what I am doing
wrong or have I miss any step?
 
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Your profit function is correct:

$$P(x)=-2x^2+48x-108$$

Now, what I would do is find the axis of symmetry (since the vertex lies on this axis)...recall that for the general quadratic:

$$y=ax^2+bx+c$$

The axis of symmetry is the line:

$$x=-\frac{b}{2a}$$

This is the arithmetic mean of the two roots. What is the axis of symmetry for your profit function?
 
Another approach would be to express your profit function in vertex form:

$$P(x)=-a(x-h)^2+k$$

where:

$$P_{\max}=k$$
 
To follow up:

The axis of symmetry is:

$$x=-\frac{48}{2(-2)}=12$$

We see that the profit function opens downward, and so the vertex is the global maximum, given by:

$$P_{\max}=P(12)=-2(12)^2+48(12)-108=12(-24+48-9)=12\cdot15=180$$

If we use the vertex form approach, we may write:

$$P(x)=-2(x^2-24x)-108=-2(x^2-24x+144)-108+2\cdot144=-2(x-12)^2+180$$

Hence:

$$P_{\max}=180$$
 
Completing the square: -2x^2 + 48x - 108= -2(x^2- 24x)- 108.
24/2= 12 and 12^2= 144. Add and subtract 144.

-2x^2+ 48x- 108= -2(x^2- 24x+ 144- 144)- 108= -2(x^2- 24x+ 144)- 288- 108

= -2(x- 12)^2- 396.

That is a parabola opening downward. For every x, it is -396 minus something. Its maximum value, -396, occurs when x= 12.
 
HallsofIvy said:
Completing the square: -2x^2 + 48x - 108= -2(x^2- 24x)- 108.
24/2= 12 and 12^2= 144. Add and subtract 144.

-2x^2+ 48x- 108= -2(x^2- 24x+ 144- 144)- 108= -2(x^2- 24x+ 144)- 288- 108

= -2(x- 12)^2- 396.

That is a parabola opening downward. For every x, it is -396 minus something. Its maximum value, -396, occurs when x= 12.

The part highlighted above in red has the wrong sign. You are subtracting 288 twice.
 
Yes, thank you for catching that.

So the parabola is $$-2(x- 12)^2+ 288- 108= -2(x- 12)^2+ 180$$.

That is a parabola, opening downward with vertex (12, 180). The maximum is 180 and occurs at x= 12.

Okay, that's a lot better- you have a positive​ profit!
 
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