How to Solve Word Problems in Calculus by Eugene Don

Author:Eugene Don
Language: eng
Format: epub
Publisher: McGraw-Hill Education
Published: 2001-04-14T16:00:00+00:00

Step 2

The distance between (x, y) and (2, 3) is

Step 3

Since the point (x, y) lies on the line,,

Step 4

Differentiating this expression is a little messy due to the presence of the radical. There is a simple trick that can be used to make the calculation easier. Instead of minimizing d, we minimize d2. Since the value of d2 is smallest when d is smallest, minimizing d2 will lead to the same point that minimizes d.

For convenience, let .

A glance at the figure should convince you that a minimum distance certainly exists. Since is the only critical number, the point is the point on the line closest to (2, 3).

EXAMPLE 14

A rectangular poster, which is to contain 50 in2 of print, must have margins of 2 in on each side and 4 in on the top and bottom. What dimensions will minimize the amount of material used?

Solution

Step 1

Although we could let x and y represent the dimensions of the poster, it turns out that labeling the inner rectangle containing the print leads to a simpler solution.

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