A rectangle is formed by bending a length of wire of length $L$ around four pegs. Calculate the area of the largest rectangle which can be formed this way (as a function of $L$).
How am I supposed to do this? If I'm interpreting the question correctly, a square would have an area of $\dfrac{1}{16}L^2$. But I don't know how to find the maximum area. I'm guessing it involves finding the stationary point of some function of $L$, but which function that might be eludes me at the moment.
Best Answer
If the rectangle is $h$ by $w$, we have the area is $A=wh$ and we have $2w+2h=L$. You solve the constraint to get $w=\frac 12(L-2h)$, and plug that into the other to get $A=\frac 12h(L-2h)$. Now take $\frac {dA}{dh}$ set it to zero, solve for $h$ and you are there. You will get the result you guessed.