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.

Explanation / Answer

f(t)=0.5t2+2t +4

df/dt =1t+2

g(x)=0.03x2+3x+20

dg/dx=0.06x+3

dx/dt =df/dt

rate at which the sales of mp3 players increases (in thousands of players per month) dg/dt=dg/dx *dx/dt

dg/dt=(0.06x+3)(t+2)

dg/dt=(0.06(0.5t2+2t +4)+3)(t+2)

dg/dt=(0.03t2+0.12t +0.24+3)(t+2)

dg/dt=(0.03t2+0.12t +3.24)(t+2)

dg/dt=0.03t3+0.18t2 +3.48t +6.48

rate at which the sales of mp3 players increases (in thousands of players per month)

=0.03t3+0.18t2 +3.48t +6.48

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