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the consumer demand curve for tissues is given by:
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q=(100-p)^2 where p is grater then or = to 0 and less then or = to 100
P is the price per case of tissues and q is the demand in weekly sales. If tissues cost $30 per case to produce, at what price should tissues be sold for the largest possible weekly profit? What will that profit be?
The question asks us to decide a price that maximizes profit. First, we must write the equation of profit and then differentiate it to find where max or min is of the profit equation.
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Profit = Income - Expense = (p - 30)*q
where p is the price and q is the demand.
by substitution, we can reduce this equation to Profit as a function of sales price.
P(p) = (p-30) * (100-p)^2 = (p-30)(10000-200p+p^2)
= p^3 - 200p^2 + 10000p - 30p^2 + 6000p - 300000
= p^3 - 230p^2 +16000p - 300000
Let's find out where the slope of this curve is zero by finding the zeroes of the first derivative.
P'(p) = 3p^2 - 460p + 16000
P'(53.blahblahblah) = 0
P''(53.blahblahblah) < 0 , relative max
Price at 53
q=(47^2)= 2209 full cases
Profit = ( 47-30)*2209 = 17*2209 = $27,553 less freight and operating expenses
We might want to perterb the number of cases to check 2210 case profit. In addition, we might negotiate a quantity order price break. :)
Erm.. one sec ill work it out
Oh wait, when i went to school, i did my own homework, that's the point of doing it, TO LEARN YOURSELF!
Good luck in life leaving school having spent it asking everyone else the questions..
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What price should tissues be sold for the largest possible weekly profit? What will that profit be?