Math Is Fun Forum

Numan

Member

Registered: 2010-04-07

Posts: 2

Find an Area from just the Perimeter (Geometry)

I've come across this question. And I cant work out the logic behind it, as for a rectangle with a certain perimeter there can be a number of possible areas for it. How would you work out all the areas for a given perimeter? Would it be factors?

Rectangle has a perimeter of 34cm. Select the answer that could be the area of the rectangle.

a)    74cm2            b) 84cm2            c) 70cm2       d) 76cm2

The answer is 70cm squared. But how is it worked out! Help pleaaaaaaaaaaaaaaaaaase!

Thank you.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Find an Area from just the Perimeter (Geometry)

If the sides of the rectangle are x and y, the perimeter is

The area is

By AM–GM,

In your example, P = 34; work out the upper bound for A from the inequality above.

Last edited by JaneFairfax (2010-04-07 01:58:03)

Numan

Member

Registered: 2010-04-07

Posts: 2

Re: Find an Area from just the Perimeter (Geometry)

Thank you very much for your reply and also your impressive solution...
Im still trying to get my head around the method, that you have used.
The number '16' in your end formula, how did that appear?

Another question/answer example:

A peremiter of a rectangle is 48cm. What could be the area of the rectangle. 258, 145, 135, 150, 172

The Answer for the area would be 135cm squared.

>>> I've tried using the method/formula and the answer I get is different.<<<<

My working out, using your formula (incorrectly I imagine!)

p = 48
A = (48 Squared) / 16
A = 144

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Find an Area from just the Perimeter (Geometry)

The 16 appeared because Jane multiplied both parts of the fraction by 4. That way she had an appearance of 2(x+y) on the top, which could be replaced with P.

You've used the formula correctly in the other question too, but remember that the formula tells you the area's upper bound. 135 is the right answer because it is less than 144, and so *could* be the area of the rectangle.
All the other options are above the upper bound and so are impossible choices.

By realising that possible areas depend only on an upper bound, you can actually save yourself quite a bit of work. With any question like this, the answer will always be the lowest option.

---

Why did the vector cross the road?
It wanted to be normal.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Find an Area from just the Perimeter (Geometry)

Hi Numan;

The fastest way to think about this is to remember that the rectangle with the biggest area for a given perimeter is a square: ( this has been proven so many times it is common knowledge )

(34 / 4 ) ^ 2 = 289 / 4 = 72.25, is the largest possible area with perimeter 34, so only c is possible.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

delexus

Guest

Re: Find an Area from just the Perimeter (Geometry)

rectangle 3y+2 
9y-1

find y if P=74cm to the secound power

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Find an Area from just the Perimeter (Geometry)

Hi delexus;

I am not sure I understand your question. What is to the second power?

If two sides of you rectangle are  3y+2 and  9y-1. With P = 74 cm then you can find y easily.

2(3y+2)+ 2(9y-1) = 74

6y + 4 + 18y - 2 = 74 Clean it up a little:

24y + 2 = 74

y = 3. The sides are 11 cm and 26 cm. The area is 11 * 26 = 286 cm^2

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.