Find the maximum area for the given perimeter of a rectangle?

Question

State the length and width of the rectangle.

25mm

18 yards

42 cm

Thanks for your help and please tell me how you do it.

Answer 1

Ok, here's how you do it.

Let's say that l will be the length, w will be the width, A will be the area and P will be the perimeter.

you know that in a rectangle, the area will be

A=wl

and the perimeter will be

P = 2w+2l

solving the perimeter equation for l:

l = (P-2w)/2

substituting in the area equation

A = w(P-2w)/2

doing some algebra

A = (Pw-2w^2)/2

ok, now you know that the maximum point of a function will be when it's slope equals to zero. And for the deffinition of a derivative, you know that it's the slope of a function, so in order to get the maximum area you have to derive it for w like this

dA/dw = (P-4w)/2

making dA = 0

(P-4w)/2 = 0

and solving for w

w = P/4

substituting in the l equation

l = (P-2*P/4)/2 = P/4

from this you can say that the maximum area of a rectangle will be when it's width and length are the same, which means it will be when the rectangle is a square. So the maximum area will be:

A = (P/4)(P/4) = P^2/16

and now that you have these equations the rest is easy.

For the 25 mm

w = l = 25mm/4

A = (25mm)^2/16 = 625/16 mm^2

For the 18 yards

w = l = 18yrds/4 = 9/2 yrds

A = (18yrds)^2/16 = 81/4 yrds^2

and finally for the 42 cm

w = l = 42cm/4 = 21/2 cm

A = (42cm)^2/16 = 441/4 cm^2

hope this helps.