A company estimates that the demand for their product (in millions) depends on the price (in dollars) according to the equation q=sqrt(4570-p^2).

Question

A company estimates that the demand for their product (in millions) depends on the price (in dollars) according to the equation q=sqrt(4570-p^2). The price is currently $35.00, but lumber shortages are driving it up at a rate of $0.127/month. Find the rate at which demand is currently changing.

Answer 1

If you're having trouble taking the derivative, maybe this will help:

First re-write the equation in exponent form as q = (4570 - p²)^1/2

Answer 2

quantity demanded = √(4570-p²) = (4570-p²)^1/2

dq/dt = (d/dt 4570-p²) * 1/2*(4570-p²)^-1/2 = (d/dt p) * (-2p) * 1/2*(4570-p²)^-1/2

dq/dt = -p*p' / √(4570-p²)

p = $35.00, p' = $0.127/mo

dq/dt = -35*0.127/mo / √(4570-35²) = -0.0769 / mo

Quantity demanded changes at a rate of -0.0769 million units / mo

(That is to say, it decreases by 76.9 thousand units per month)

Answer 3

p = current price

q = demand in millons of units

Need to find dq/dt when p=$35 and dp/dt = 0.127 dollars/month

q=√(4570-p²)

dq/dt = (-p/√(4570-p²)) dp/dt = (-35/√(4570-1225))(0.127) ≈-.077 million units/month