Answer to Question #154588 in Calculus for Phyroe

Question

A triangle has a base 12 ft long and an altitude 8 ft high. Find the area of the largest rectangle that can be inscribed in the triangle so that the base of the rectangle falls in the base of the triangle.

Answer 1

Solution

Let |AC|=12, |BH|=8, |GF|=|DE|=w, |DG|=|EF|=h.

Triangles ABC and DBE are similar.

So |DE|/|AC| = |BK|/|BH| => w/12 = (8-h)/8 => w = 3*(8-h)/2

Rectangle area S = w*h = 3*h*(8-h)/2 = 3*(8h-h²)/2

Necessary Condition for an Extremum of this function on h is

dS/dh = 0 => 8-2*h=0 => h = 4 ft

w = 3*(8-h)/2 = 12/2 = 6 ft

S = w*h = 24 ft²

Answer

S = 24 ft²