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Example 1: Cost Optimization

A woman wishes to rent a house. If she lives x miles from her work, her transportation cost will be cx dollars per year, while her rent will be 25c/(x + 1) dollars per year. How far should she live from work to minimize her rent and transportation expenses?

Let y be her expenses in dollars per year. Then

03_continuous_functions-209.gif

The problem is to find the minimum value of y in the interval 0 ≤ x < ∞.

Step 1

03_continuous_functions-210.gif

Step 2

To find x such that dy/dx = 0 we set dy/dx = 0 and solve for x.

03_continuous_functions-211.gif , (x + 1)2 = 25, x + 1 = ±5.

Then x = 4 or x = -6. We reject x = -6 because 0 ≤ x. The only interior critical point is x = 4.

Step 3

We use the Direct Test.

At x = 0,

y = c · 0 + 25c/(0 + 1) = 25c.

At x = 4,

v = 4c + 25c/(4 + 1) = 9c.

At x = 9,

y = 9c + 25c/(9 + 1) = 11.5c.

CONCLUSION y has its minimum at x = 4 miles. So the woman should live four miles from work. (See Figure 3.6.1.)

03_continuous_functions-212.gif

Figure 3.6.1

Home Continuous Functions Maxima and Minima - Applications Examples Example 1: Cost Optimization

Last Update: 2006-11-25