Suppose the wholesale price of a certain brand of medium-sized eggs p (in dollars/carton) is related to the weekly supply x (in thousands of cartons) by the following equation.$ 625p^2 - x^2 =100$If 35000 cartons of eggs are available at the beginning of a certain week and the price is falling at the rate of 5¢/carton/week, at what rate is the supply changing? (Round your answer to the nearest whole number.) (Hint: To find the value of p when x = 35, solve the supply equation for p when x = 35.)___ cartons per week?

Accepted Solution

Answer:1300Step-by-step explanation:We are given x = 35[tex]\frac{dp}{dt}=-0.05[/tex]  [rate of change of price with time]We want to find [tex]\frac{dx}{dt}[/tex], which is the rate at which supply is changing.First, we need to implicitly differentiate the equation given:[tex]625p^2-x^2=100\\1250p\frac{dp}{dt}-2x\frac{dx}{dt}=0[/tex]Here, we have what we need except for p. We find p when x = 35:[tex]625p^2 - x^2 =100\\625p^2 - (35)^2 =100\\625p^2=1325\\p=1.46[/tex]Now, find dx/dt:[tex]1250(1.46)(-0.05)-2(35)\frac{dx}{dt}=0\\-91.25-70\frac{dx}{dt}=0\\\frac{dx}{dt}=-1.30[/tex]In thousands, 1.30 * 1000 = 1300 cartons per week decrease