What is the largest area of an isosceles triangle with the perimeter 1 cm?
Hi
Let a be the equal length of 2 sides and b the length of the 3rd side.
The perimeter is p = 1 = 2a+b
The are is given by the formula $\displaystyle S = \sqrt{\frac{p}{2}\left(\frac{p}{2}-a\right)\left(\frac{p}{2}-a\right)\left(\frac{p}{2}-b\right)}$
$\displaystyle S = \sqrt{\frac{1}{2}\left(\frac{1}{2}-\frac{1-b}{2}\right)\left(\frac{1}{2}-\frac{1-b}{2}\right)\left(\frac{1}{2}-b\right)}$
$\displaystyle S = \sqrt{\frac{1}{2}\:\frac{b}{2}\:\frac{b}{2}\:\frac {1-2b}{2}}$
$\displaystyle S = \frac{b}{4}\:\sqrt{1-2b}$
Set the derivative of S to 0 to find b_max then S_max