SOLUTION: what is the geatest possible area of a rectangle if its perimeter is 30 cm

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Question 214015: what is the geatest possible area of a rectangle if its perimeter is 30 cm
Found 3 solutions by rfer, drj, josmiceli:
Answer by rfer(16322) About Me  (Show Source):
You can put this solution on YOUR website!
A square is the largest area
A=7.5*7.5
A=56.25 sq cm

Answer by drj(1380) About Me  (Show Source):
You can put this solution on YOUR website!
What is the greatest possible area of a rectangle if its perimeter is 30 cm.

The greatest possible area is when the sides are equal or a square and the perimeter is adding up the lengths of all four sides in this case.

So let s be the side of the square.

Then Perimeter P=4s or s=P/4=30/4=7.5 cm.

The Area A=s%5E2=7.5%5E2 or 56.25 square centimeters.

I hope the above steps were helpful.

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Good luck in your studies!

Respectfully,
Dr J




Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
If the sides are h and w,
then 2h+%2B+2w+=+30
2w+=+30+-+2h
w+=+%281%2F2%29%2A%2830+-+2h%29
A+=+w%2Ah
A+=+%281%2F2%29%2A%2830+-+2h%29%2Ah
A+=+%281%2F2%29%2A%2830h+-+2h%5E2%29
A+=+-h%5E2+%2B+15h
The - sign tells me this is a parabola with
a peak at the top, or a maximum
The maximum occurs exactly between the
roots, or at }-b%2F%282a%29
b+=+15
a+=+-1
h%5Bmax%5D+=+-b%2F%282a%29
h%5Bmax%5D+=+-15%2F-2
h%5Bmax%5D+=+7.5
and
w+=+%281%2F2%29%2A%2830+-+2h%29
w+=+%281%2F2%29%2A%2830+-+15%29
w+=+7.5
A 7.5 x 7.5 rectangle, or a square, has the max area
which is7.5%5E2+=+56.25 cm2
You can prove this by changing the square slightly
and keep the perimeter the same, say
7.4 x 7.6
perimeter = 2%2A7.4+%2B+2%2A7.6+=+14.8+%2B+15.2
14.8+%2B+15.2+=+30
A+=+7.4%2A7.6
A+=+56.24 (just a little smaller than 56.25)