Page 21 - How To Solve Word Problems In Calculus

Finally, we substitute this expression for y into the area
equation obtained in step 2.

A = xy

3
A = x 750 − x
2
3
A(x) = 750x − x 2
2

Mathematically, the domain of A(x) is the set of all real
numbers. However, in this problem, as with all geometry prob-
lems, negative dimensions are unrealistic. Although x = 0 may
appear to be unrealistic as well, we generally allow a rectangle
of zero width or length with the understanding that its area
is 0. Such a rectangle is called a degenerate rectangle. Since the
perimeter is fixed, y gets smaller as x gets larger so the largest
value of x occurs when y = 0.

3x + 2y = 1500

3x = 1500 (y = 0)
x = 500

The function describing the area of the farmer’s pen is

3
A(x) = 750x − x 2 0 ≤ x ≤ 500
2
EXAMPLE 6

A piece of wire 12 inches long is to be used to form a square
and/or a circle. Determine a function that expresses the com-
bined area of both figures. What is its domain?

Solution
Step1
Let x be the side of the square and r the radius of the
circle. We shall express the area as a function of x.

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