A triangle has two equal sides 4 inches long. what is the length of the third side (the base) if the triangle is to have maximum area?

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Answer 1 · 2015-11-01T23:57:04

#4sqrt(2)# inches

Suppose the base is of length #x# inches.

Using Pythgoras, the height of the triangle is:

#sqrt(4^2-(x/2)^2)#

So the area of the triangle is:

#a(x) = 1/2 xx "base" xx "height" = 1/2 x sqrt(4^2-(x/2)^2)#

then #16(a(x))^2 = 64x^2-x^4= 1024-(x^2-32)^2#

This will have its maximum value when #x^2-32 = 0#, that is when #x = +-sqrt(32) = +-4sqrt(2)#.

Since we are dealing with lengths, which must be non-negative, we can deduce that #x = 4sqrt(2)#