If the profit equation for the monthly sales of tile sets is P = -x^2 + 56x - 300, how do I find the number of tiles that yeilds the most profit.

x=number of tile sets. The amount of money for costs each month is set at $300.
I saw an answer as 28 tile sets would yeild the highest profit but I don't understand why that is the answer, if it is.
Could you help me? Thanks so much.

x^2 -56 x = -p - 300

complete the square to find vertex of parabola

x^2 -56 x + 784 = -p +484

(x-28)^2 = - (p-484)
so
vertex (max of p) is at (28,484)

Damon,

Why does this explain the highest yeild?

It sounds like what you did was on a graph and you found the highest points

-4x^3 + 24x^2 – 24x + 4 by x^2 – 5x + 1

Of course! I'd be happy to help you understand how to find the number of tiles that yields the most profit.

To find the number of tile sets that yields the highest profit, you need to find the maximum value of the profit equation P = -x^2 + 56x - 300. In this equation, x represents the number of tile sets.

One way to find the maximum value is by using calculus. To do this, we need to take the derivative of the profit equation with respect to x and set it equal to zero. So let's go through the steps:

First, take the derivative of the profit equation with respect to x. The derivative of -x^2 is -2x, the derivative of 56x is 56, and the derivative of -300 is 0.

So the derivative of P with respect to x is P' = -2x + 56.

Next, set the derivative equal to zero and solve for x:
-2x + 56 = 0

To isolate x, subtract 56 from both sides:
-2x = -56

Now divide both sides by -2 to solve for x:
x = 28

So according to the calculation, x = 28 tile sets would yield the highest profit. This means that producing and selling 28 tile sets would result in the maximum profit.

To understand why 28 is the answer, let's take a closer look at the equation. The profit equation P = -x^2 + 56x - 300 is quadratic, and the coefficient of the x^2 term is negative, indicating a downward opening parabola.

This means that the profit initially increases as the number of tile sets (x) increases up to a certain point, and then starts decreasing. The maximum point of the parabola represents the highest profit. By setting the derivative equal to zero, we found that x = 28 is the value where the slope of the profit curve changes from positive to negative, indicating the maximum profit.

Therefore, producing and selling 28 tile sets would result in the highest profit according to the given profit equation.

I hope this explanation helps! Let me know if you have any further questions.