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Calculus IChapter 4: Application of DerivativesSection 4-14

Business Applications — Question 5

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Question 5

A company produces and sells xx units of a product per week. The weekly revenue and cost functions (in thousands of dollars) are: R(x)=200x−2x2,C(x)=50x+300R(x) = 200x - 2x^2, \quad C(x) = 50x + 300

  • (a) Find the profit function P(x)P(x).

  • (b) Determine the number of units that maximizes the profit.

  • (c) Compute the maximum profit.

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Question 5 - Solution

P(x)=R(x)−C(x)=−2x2+150x−300.P(x)=R(x)-C(x)=-2x^2+150x-300.

In the continuous model, P′(x)=−4x+150=0P'(x)=-4x+150=0 gives x=37.5x=37.5 and P″(x)=−4<0P''(x)=-4<0.

The continuous maximum is 2512.52512.5 thousand dollars.

For whole units, compare x=37x=37 and x=38x=38, the integers nearest the vertex:

P(37)=P(38)=2512.P(37)=P(38)=2512.

Thus the feasible maximum is

37 or 38 units,2512 thousand dollars.\boxed{37\text{ or }38\text{ units},\qquad2512\text{ thousand dollars}.}

Original worksheet page 2: question and worked solution for 4-14-005

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