Find the revenue

Question

Mobile phone manufacture will be releasing a new version of their smart phone model. They

determined that t months after the new phone introduced to the market;

The manufacturing facility can produce x(t) hundred units.

The price of the unit p(t) will be changed by the time as well.

Given

t^2+3t hundreds of units produced where t is the month since the release time

p(t) = -2t^(3/2) + 30 dollars per unit.

The revenue R(t) = x(t) p(t)

At what rate is the revenue changing with respect to the time after 4 months?

Is the revenue increasing or decreasing at this time?

My answer is -14% revenue

Answer 1

Since R(t) = x(t)p(t) the product rule is applicable

So R'(t) = x'(t)p(t) + x(t)p'(t)

x(t) = t² + 3t

x'(t) = 2t + 3

p(t) = -2t^3/2 + 30

p'(t) = -3t^1/2

R'(t) = (2t + 3)(-2t^3/2 + 30) + (t² + 3t)(-3t^1/2)

R'(4) = (2•4 + 3)(-2(4)^3/2 + 30) + ((4)² + 3(4))(-3(4)^1/2)

R'(4) = (11)(14) + (28)(-6)

R'(4) = 154 - 168 = -14

Since the units of Revenue are dollars and the units of time (t) is months, then R' has units of $/mo (not %).

Since the slope of the Revenue vs time curve is negative (-$14/mo), then the Revenue is decreasing at t = 4 months.