Question
In: Math
Suppose the wholesale price of a certain brand of medium-sized eggs p (in dollars per carton)...
Suppose the wholesale price of a certain brand of medium-sized eggs p (in dollars per carton) is related to the weekly supply x (in thousands of cartons) by the equation
625?2 −?2 = 100
If 25,000 cartons of eggs are available at the beginning of a certain week, and
(a) if the price is falling at the rate of 2¢/carton/week, at what rate is the weekly supply changing?
(b) if the weekly supply is falling at the rate of 1,000 cartons/week, at what rate is the wholesale price changing? Give your answer to the nearest tenth of a cent.
Solutions
Expert Solution
The wholesale price p(in dollars per carton) is related to the weekly supply x(in thousands of cartons) by the equation
25,000 cartons of eggs are available at the beginning of a certain week which means x = 25.
The wholesale price p when x = 25 is
(a)
The price is falling at the rate of 2¢/carton/week which means
Differentiating the price-demand equation w.r.t. time t(in weeks), we get
The rate of change of weekly supply in the given condition is
Hence, the weekly supply is falling at the rate of 538.5
539 cartons/week.
(b)
The weekly supply is falling at the rate of 1,000 cartons/week which means
Differentiating the price-demand equation w.r.t. time t(in weeks), we get
The rate of change of wholesale price in the given condition is
Hence, the wholesale price is falling at the rate of 0.037 $/carton/week.
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