Question

a rectangle of perimeter 100 units has the dimensions shown. Its area is given by the function A = w(50 - w). What is the GREATEST area such a rectangle can have? Please explain all steps written, there are parts such as 50W -W^2 and it ends up turning into the next part how?

Answer 1

If A=w(50-w)

A=50w-w^2

dA/dw=50-2w

d2A/dw2=-2

Since the acceleration is a constant negative, that means that when velocity, dA/dw=0, it is at an absolute maximum for A(w)...

dA/d2=0 only when 50=2w, w=25

So as the case with any rectangle, the perfect square will enclose the greatest area possible with respect to a given amount of material to enclose that area...

So the greatest area occurs when W=L=25 in this case:

A(25)=50w-w^2

Area maximum is thus:

Amax=50(25)-(25)^2=625 u^2