Question

When toasters are sold for p dollars a piece, local consumers will buy D(p)=57,600/p toasters a month. It is estimated that t months from now,the price of the toasters will be p(t)=0.03t^3/2 + 22.08 dollars. Compute the rate at which the monthly demand for the toasters will be changing with respect to time 16 months from now.
I get -75.47, it will decrease. Is that correct?

Answer 1

I get dD/dt = dD/dp * dp/dt

-57600/(24^2) * .03(3/2)(4)

I get it at: -18. Demand is dropping at a rate of 18 units at 16 months.

But I could be wrong :)

Answer 2

where do you get 24^2

Answer 3

p(16) = .03 (sqrt(16^3)) + 22.08
.03(16*4) + 22.08
.03(64) + 22.08
1.92 + 22.08 = $24.00 at 16 months

Answer 4

dD/dp = -57600/p^2; p=24 at 16 months.

Answer 5

for the 57600/p^2 p=24 at 16 months
shouldn't that be the derivative of 57699/p which I think would be -115200p^-2?

Answer 6

Quotient rule:

top: (BT')-(B'T)
bottom: B^2

T=57600; T' = 0
B=p ; B'=1

top: (p*0) - (1*57600)
bottom: p^2

The change in demand (D) with respect to price (p):

(dD/dp) = -57600/p^2