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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?

Answer 1

William Abdalla

#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)#

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Answer 2

Charles Anctil

To maximize the area of the triangle, the third side (base) should be twice the length of the equal sides. Therefore, the length of the third side (base) would be 8 inches.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.