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Calculus IChapter 4: Application of DerivativesSection 4-14

Business Applications — Question 4

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Question 4

The demand for a product is modeled by p(x)=100−2x,p(x)=100-2x, where p(x)p(x) is the price per unit in dollars and xx is the number of units sold in hundreds. The total cost of producing 100x100x units is C(x)=20x+200dollars.C(x)=20x+200\quad\text{dollars}. Require x≥0x\geq0 and p(x)≥0p(x)\geq0.

  • (a) Find the revenue and profit functions.

  • (b) Find the production level and maximum profit in the continuous model.

  • (c) Find the best whole-number quantity of units and its profit.

Original worksheet page 1: question and worked solution for 4-14-004
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Question 4 - Solution

(a) Revenue and profit

The number sold is 100x100x, so revenue is 100xp(x)100x\,p(x):

R(x)=10000x−200x2,P(x)=−200x2+9980x−200.R(x)=10000x-200x^2,\qquad P(x)=-200x^2+9980x-200.

P′(x)=−400x+9980,P′′(x)=−400<0.P\prime(x)=-400x+9980,\qquad P\prime\prime(x)=-400<0.

(b) Continuous maximum

The continuous maximum occurs at

x=9980400≈24.950000,Pmax≈$124,300.50.\boxed{x=\frac{9980}{400}\approx 24.950000,\quad P_{\max}\approx\$124,300.50.}

(c) Whole units

This represents 2495.0000002495.000000 units. Comparing the adjacent whole-unit counts gives 2495\boxed{2495} units, with profit $124,300.50\boxed{\$124,300.50}. The price is nonnegative at these levels.

Original worksheet page 2: question and worked solution for 4-14-004

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