The sum of the legs of a right triangle is 36 cm. For what lengths of the sides will the square of the hypotenuse be a minimum?

1 Answer
MeneerNask
Jan 8, 2016

We can do this in two ways: by lateral thinking or in the robust mathematical way,

Explanation:

Let's do the first way , assuming both legs are 18 cm. Then the square of the hypotenuse will be #18^2+18^2=648#
If we change this to #17harr19# it will be #650#
Even #10harr26# will give a greater number: #686#
And #1harr35# will lead to #1226#

The mathematical way:
If one leg is #a# then the other one is #36-a#
The square of the hypotenuse is then:
#a^2+(36-a)^2=a^2+1296-72a+a^2#
Now we have to find the minimum of:
#2a^2-72a+1296# by setting the derivative to 0:
#4a-72=0->4a=72->a=18#

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