Homework Help Please

Question

A company has found that the number of items sold, x, depends upon the price, p at which they're sold, according the equation x=30400−2p^2

The price has been increasing by $2 per month. Find the rate at which revenue is changing when the price is $70. Note: the answer might be negative

Revenue =(Price)⋅(Number Sold)

 

______________ dollars per month

Answer 1

when p=70, x= 30400-2(70)^2 = 1000.

multiply 2 times 1000 = $2000 per month gain in revenue

yet any increase in price means x goes down, quantity purchased decreases as price increases.

at some point any price increase means a decrease in revenue, with R'<0.

this is not a linear demand function though which is the usual convention. so unit price elasticity of

demand is at higher price than with a linear demand equation.

Answer 2

x = number of items sold

p = price

R = revenue

rate of change of price (dp/dt) = $2 per month

x = 30400 - 2p²

R = p * x = p (30400 - 2p²) = 30400p - 2p³

Take the derivative of R to find the rate of change of revenue (dR/dt):

dR/dt = d/dt [30400p - 2p³] = 30400(dp/dt) - 6p²(dp/dt)

Substitute p = $70 and dp/dt = 2:

dR/dt = 30400 (2) - 6(70)² (2) = $2000 per month