Question

An electronics store is deciding how to price one of its products. The equation P=-25d^2+500d predicts the total profit P as a function of the product's price in dollars d. What price will produce the highest total profit?

Answer 1

1. The function calculating the profit in terms of the price in dollars d, is the quadratic function $P(d)=- 25d^{2}+500d$, whose graph is a concave parabola, that is a parabola which opens downwards, because the coefficent of $d^{2}$ is negative. 

2. The function P takes its maximum value at the vertex, so we want to find the x coordinate of the vertex. 

3. $P(d)=- 25d^{2}+500d=-25d(d-20)$ is the factorized form of the function P, which clearly gives the roots (solutions) 0 and 20 of P(d)=0, which are the x intercepts of the parabola. 

4. The axis of symmetry, which contains the vertex, is the vertical line passing through the point equidistant to 0 and 20, that is 10.

5. Remark : The vertex is (10, P(10))=(10, -25*10(10-20))=(10, 2500)

So for price d=10 dollars, the (expected) Profit is 2500 dollars.

6. Remark 2. The x coordinate is also found by the formula -b/2a, where b is the coefficient of the linear term, and a the coefficient of the quadratic term of a quadratic expression. So in our case -b/2a=-500/2(-25)=-500/-50=10