Max Net Profit

Calculus Level pending

XYZ, Inc. uses the functions R ( x ) R(x) and E ( x ) E(x) to model their total revenue (R) and cost of selling widgets (E) at various price levels. If maximal revenue and maximal net profit are achieved at two different price levels, how much less will total revenue be if the widgets are sold at a price that will maximize net profit?

R ( x ) = − 200 x 2 + 7000 x R(x) = -200x^2+7000x
E ( x ) = − 2400 x + 84000 E(x) = -2400x+84000


The answer is 7200.

Justin Malme by Justin Malme , 36, USA

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1 solution

Justin Malme
Jun 9, 2016

If R ( x ) = − 200 x 2 + 7000 x R(x) = -200x^2+7000x , we can calculate the derivative and get R ′ ( x ) = − 400 x + 7000 R'(x) = -400x+7000 .
Solving for x x in R ′ ( x ) = 0 R'(x) = 0 , we see that x = 17.5 x=17.5 .
Therefore R ( 17.5 ) = $ 61 , 250 R(17.5) = \$61,250 , our maximal Revenue.

To find max net profit, we need to create a new function P ( x ) P(x) to model net profit.
P ( x ) = R ( x ) − E ( x ) = − 200 x 2 + 7000 x − ( − 2400 x + 84000 ) = − 200 x 2 + 9400 x − 84000 P(x) = R(x) - E(x) = -200x^2+7000x - (-2400x+84000) = -200x^2+9400x-84000 .
Calculating the derivative of P ( x ) P(x) , we get P ′ ( x ) = − 400 x + 9400 P'(x) = -400x+9400 .
Solving for x x in P ′ ( x ) = 0 P'(x) = 0 , we see that x = 23.5 x=23.5 , the price point that achieves maximal net profit.
Therefore, revenue at the price point of max net profit will be R ( 23.5 ) = $ 54 , 050 R(23.5) = \$54,050 .

R ( 17.5 ) − R ( 23.5 ) = $ 61 , 250 − $ 54 , 050 = $ 7 , 200 R(17.5) - R(23.5) = \$61,250 - \$54,050 = \$7,200

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