Page 121 - How To Solve Word Problems In Calculus

Since A(x) is zero at each endpoint of the interval [0, 200],
the absolute maximum area occurs at x = 100. The corresponding
value of y is 400 − 2x = 400 − 200 = 200 ft. The maximum area
2
xy = 20,000 ft .
3. Label the legs of the triangle x and y and the hypotenuse z.

z
y

x

1
The area of the right triangle is A = xy. There are two
2
relationships between x, y, and z: x + y + z = 20 and

2
2
z = x + y . These may be combined to eliminate z.

2
2
x + y + x + y = 20
We begin by simplifying this equation. To solve for y, we rewrite the
equation leaving only the radical on one side.

2
2
x + y = 20 − (x + y)
To eliminate the radical, square both sides of the equation. Then
simplify.

2
2
x + y = 400 − 40(x + y) + (x + y) 2
2
2
2
x + y = 400 − 40x − 40y + x + 2xy + y 2
0 = 400 − 40x − 40y + 2xy
To solve for y in terms of x, we bring all terms involving y to the
left side of the equation. Terms not involving y remain on the
right.

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