SOLUTION: A triangle has two equal sides and. Third side. Determine the ratio between the sides a and b to enclose the maximum area for a given total length of the sides?

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Question 1072946: A triangle has two equal sides and. Third side. Determine the ratio between the sides a and b to enclose the maximum area for a given total length of the sides?
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Look at an isosceles triangle with sides, m, m, and 2n.
So that
m=a
n=b%2F2
The perimeter is then,
P=m%2Bm%2B2n
P=2m%2B2n
The perimeter is set to a constant length, 2L.
2m%2B2n=2L
m%2Bn=L
n=L-m
The area would then be,
A=%281%2F2%29%28Base%29%28Height%29
A=%281%2F2%29n%28sqrt%28m%5E2-n%5E2%29%29
Substituting for n,
A=%281%2F2%29%28L-m%29%28sqrt%28m%5E2-%28L-m%29%5E2%29%29
A=%281%2F2%29%28L-m%29%28sqrt%282mL-L%5E2%29%29
So to find the maximum area, take the derivative of A with respect to a and set it equal to zero.
dA%2Fda=%28L%282L-3m%29%29%2F%282%28sqrt%282mL-L%5E2%29%29%29
%28L%282L-3m%29%29=0
One solution,
2L-3m=0
3m=2L
m=%282%2F3%29L
and
n=L-%282%2F3%29L
n=%281%2F3%29L
So in terms of a and b,
m=a
and
n=b%2F2
So substituting,
a=%282%2F3%29L
and
b%2F2=%281%2F3%29L
b=%282%2F3%29L
So the ratio is,
a%2Fb=%28%282%2F3%29L%29%2F%28%282%2F3%29L%29
highlight%28a%2Fb=1%29