Need Help on Algebra 2

Question

The per-unit price p (in dollars) of a popular toy when a units (in thousands) are produced is modeled by the function p= 12 - 0.025x. The revenue (in thousands of dollars) is the product of the price per unit and the number of units (in thousands) produced

So the revenue function is given by r(x) = xp, or rather:

r(x) = x (12- 0.025x)

According to this function, what is the maximum possible revenue? Type in the number of dollars (without a dollar sign), Be careful, r(x) gives thousands of dollars, not dollars.

Answer 1

r(x) = x (12- 0.025x)

If you multiply through by the x, you'll see that this is an inverted parabola (vertex at the top). The maximum revenue will occur at the vertex. Note that r(x) = 0 when x = 0 and again when 12 - 0.025x = 0:

12 - 0.025x = 0

12 = 0.025x

12/(0.025) = x

480 = x

So r(x) = 0 when x = 0 and x = 480. When you have two points on a parabola with the same y value, the x coordinate of the vertex is exactly halfway between them, so the vertex lies at the point x = (480-0)/2 = 240. To find the max revenue, plug x = 240 into the r(x) equation. Remember the answer you get is in thousands of dollars, not dollars. To convert to dollars, multiply the answer by 1000.