A manufacturer is willing to make x thousand six packs of beer per week availabl

Question

A manufacturer is willing to make x thousand six packs of beer per week available in the market place when the wholesale price is p dollars per six pack. It is known the relationship between x and p is governed by the supply equation x^2-3xp+2p^2=7. Use calculus to find how fast with respect to time the supply of beer is changing when the price per six pack is $6, the quantity supplied is 5000 six packs and the wholesale price per six pack is increasing at a rate of 20 cents per six pack per week? Please show work in detail so I can follow. Thank you so much!

Explanation / Answer

x^2-3xp+2p^2=7 diff. wrt to time 2xdx/dt - 3pdx/dt - 3xdp/dt + 4pdp/dt = 0 dx/dt = (3x-4p)(dp/dt)/(2x-3p) given that, p = 6 x = 5000 dp/dt = .2 putting these values, dx/dp = (3*5 -4*6)(.2)/(2*5-3*6) dx/dp = .225 thousand six packs/ week dx/dp = 225 six packs/week

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