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 February 21st, 2012, 11:22 PM #1 Senior Member   Joined: Jan 2012 Posts: 100 Thanks: 0 Differentials and Linear Approximation Hey guys, I was working hard lately and understanding the new material my teacher had handed out, but I came across this one question and I am stuck. Can anyone please help me? Question: Suppose the wholesale price of a certain brand of medium-sized eggs $p$ is related to the weekly supply $x$ by the equation $625p^2-x^2=100$ If 25000 cartons of eggs are available at the beginning of a certain week and the price is falling at the rate of $2c$/carton/week, at what rate is the supply falling? We are given a hint: to find value of $p$ when $x=25$, solve the supply equation for $p$ when $x=25$. Please help, looking forward to your responses.
 February 21st, 2012, 11:29 PM #2 Senior Member   Joined: Jan 2012 Posts: 100 Thanks: 0 Re: Differentials and Linear Approximation I forgot to include my answer: I needed to rate p wrt t, so i took the whole thing and found $d/dt$ 1250p(dp/dt)-2x(dx/dt)=0 dp/dt=[2x(dx/dt)]/1250p I plugged in 25 for original formula here: 625p^2-(25)^2=100 which came out to $p=sqrt725/625)$ I plugged in that value of P into the other equation: dp/dt=[2x(dx/dt)]/1250p=[2(25)(-1)]/1250(sqrt(725/625))=-$0.37/week = -$3.7 cents/week is that correct? and if possible can you show me using latex, i would have submitted mine with latex, but i was tired of fighting latex.
 February 21st, 2012, 11:52 PM #3 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs Re: Differentials and Linear Approximation I take it that x represents thousands of cartons. You want to find $\frac{dx}{dt}$. Differentiating the equation relating p and $x$ with respect to time t gives: $1250p\frac{dp}{dt}-2x\frac{dx}{dt}=0$ $\frac{dx}{dt}=\frac{625p}{x}\cdot\frac{dp}{dt}$ Now, we are given $\frac{dp}{dt}=-2$ and when x = 25, $p=\frac{\sqrt{29}}{5}$ and so: $\frac{dx}{dt}=5\sqrt{29}(-2)=-10\sqrt{29}\text{ \frac{cartons}{week}}$ I missed the part where we use a differential to get an approximation.

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