Thursday, September 22, 2011

How to solve this another calculus problem?

The demand equation for a particular toy is p^2x = 5000, where x toys are demanded per month when p dollars is the price per toy. It is expected that in t months, where t is an element of [0, 6], the price of the toy will be p dollars, where 20p = t^2 + 7t + 100. What is the anticipated rate of change of the demand with respect to time in 5 months? Do not express x in terms of t, but use the chain rule. Also please show on how it is done.How to solve this another calculus problem?
Rate of change of demand = dx/dt

Structure as chain rule as question asks:

dx/dt = (dx/dp) . (dp/dt)

Calculate dx/dp:

2x lnp = ln5000

2(dx/dp) lnp + 2x(1/p) = 0

(dx/dp) lnp + x/p = 0

dx/dp = (-x/p) / lnp = -x / (p lnp)

dp/dt = 2t + 7

dx/dt = -x(2t + 7) / (p lnp)

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