R=-1/728t^2 + 9t

Question

Find the maximum revenue.

Answer 1

Noticing that the highest order in this equation is 2 (t²), we have a parabola. Since the coefficient on the highest order term is negative, it opens downward, and has a maximum a the vertex.

If we have the general form of a parabola, y=ax²+bx+c we can find the vertex (h,k) if we get it into the form y=a(x-h)²+k

Using the quadratic formula, we can find h as -b/(2a) and k as (4ac-b²)/(4a)

In our equation, a =-1/728, b=9, and c=0.

h would give us the time at max revenue, and k would be the max revenue, so we don't need to calculate h, just k:

k=-(9)²/(4*-1/728)

max revenue = 14742.

Answer 2

This is a parabolic curve. Revenue increases, and then decreases until reaching zero. Halfway between reaching zero is our maximum revenue. If we solve for the total time t until reaching zero, and halve that, we will obtain our maximum revenue.

First solving for t

R = -1/728t² + 9t

0 = -1/728t² + 9t

1/728t² = 9t

1/728t = 9

t = 9 x 728

t = 6552

We plug half this value into our original equation to get maximum revenue

R = -1/728(6552/2)² + 9(6552/2)

R = 14742

Hope that helps!