Page 67 - Calc for the Clueless

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The area A = 2xy. Since the point (x,y) is on the curve, y = 12 - x . Sooooo...

only +2 is used—it is a length

The area is 2xy = 2(2)(8) = 32.

Example 10—

A rectangle is to be inscribed in a right triangle of sides 6, 8, 10 so that two of the sides are on the triangle. Find
the rectangle of maximum area.

We must note three things in this problem. First, we must set up the triangle in terms of an x-y coordinate
system, with the legs on the axes. Second, we note that point B, wherever it is, is represented by the point (x,y),
so that the area of the rectangle A = xy. The third thing we must note is that the way to relate x to y is the
similar triangles BCD and ACE. Since EC = 6, and ED = x, DC = 6 - x. The proportion we get is

Solving for y, we get

The rectangle of maximum area is 4(3) = 12.

Example 11—