The number of thousand mp3 players produced per month by a certain manufacturer

Question

The number of thousand mp3 players produced per month by a certain manufacturer t months after the start of the year 2004 can be described by the function

As they sell more players, the company's profit increases both because they are selling more players and because economies of scale mean they can produce each player more cheaply.

The manufacturer's total profit (in thousands of dollars) when x thousand mp3 players are made is given by the function

We assume that the company sells all of the mp3 players that it produces.

Calculate the rate at which the sales of mp3 players increases (in thousands of players per month) in terms of t.   

Rate of increase :  thousand players per month.

Calculate the rate at which the manufacturer's profit increases (in thousands of dollars per thousand mp3 players) in terms of x.

Rate of increase :  thousand dollars per thousand mp3 players.

Calculate the rate at which the manufacturer's profit increases at time t (in thousands of dollars per month).

Leave your answer in terms of t alone (x should not appear in your answer).

Rate of increase :  thousand dollars per month.

Explanation / Answer

df/dt = 0.8t + 2

dg/dx = 0.04x + 4

dg/dt = dg/dx*dx/dt

dx/dt = df/dt

dg/dt = (0.04x + 4)(0.8t + 2)
dg/dt = (0.04*(0.4t^2 + 2t + 3) + 3)(0.8t + 4)
dg/dt = (0.016t^2 + 0.08t + 3.12)(0.8t + 4) Answer

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