what is the length and width of a window with a perimeter of 25 and a area of 36

Question

This is a rectangular window, so do I use a rectangular formula for area or perimeter or no? That's why I asking a tutor because I not really sure what to do on this problem. If you could be very specific, it would help me because I am a visual learner! Thanks for all of your help! Bye

Answer 1

You need to use both--information of one will help solve the other.

Let's start by listing our formulas and what we know:

P = 2l + 2w

P = 25

A = lw

A = 36

 

25 = 2l + 2w

36 = lw

 

       L

-----------

|            |  W

|            |

|     36in²|

|            |

-----------

 

Let's try some substitution to help solve our equation:

25 = 2l + 2w

12.5 = l + w

12.5 - l = w

 

36 = l(12.5 - l)

36 = 12.5l - l²

0 = -l² + 12.5l - 36

If we divide our equation by -1, we get:

l² - 12.5l + 36 = 0

 

Factoring with that fraction isn't going to be pretty, and I'm ambivalent about completing the square, so let's try the quadratic formula:

(-b ± √(b² - 4ac)) / 2a

(12.5 ± √(12.5² - 4 * 1 * 36)) / 2 * 1

(12.5 ± √(156.25 - 144)) / 2

(12.5 ± √12.25) / 2

(12.5 ± 3.5) / 2

 

(12.5 + 3.5) / 2  -->16 / 2 = 8

(12.5 - 3.5) / 2  --> 9 / 2 = 4.5

 

Since we have two possibilities, let's try them out and see if they work:

25 = 2*8 + 2w

25 = 16 + 2w

9 = 2w

w = 4.5

 

Since 4.5 is our other possibility, then those are probably our length and width. We will test that by putting them into our area formula:

36 = 8 * 4.5

36 = 36

 

So our length really is 8 units, and our width is 4.5 units. (You did not mention if it's inches/feet/centimeters.)

Answer 2

This is a VERY HARD problem for a 7th grader. The method above is perfect, but I bet you don't know how to solve linear equations let alone quadratic equations. So I would make a table for the area of 36, then calculate the perimeter.....as follows.

 

length  *   width  =    area                  perimeter

1               36             36                     1+36+1+36=74       not right

2               18             36                     2+18+2+18= 40      not right

3               12             36                     3+12+3+12=30       not right

4                 9             36                     4+9+4+9 = 26        CLOSE, but not right

5                7.2           36                     5+7.2+5+7.2=24.4   TOO Small??????

 

So let's try something in between 4 and 5 since the perimeter associated with 4 was too big and the perimeter that goes with 5 is too small. Let's try 4.5

 

4.5             8               36                    4.5+8+4.5+8=25       PERFECT!!!!!!

 

So the dimensions of the window are 4.5 by 8.

 

Hope this helped.

Answer 3

 

P=2l+2w

25=2l+2w

A=lw

36=lw

36/w=l

25=2l+2w

substitute

25=2(36/w)+2w

25=(72/w)+2w

multiply both sides by w

25w=w(72/w)+w(2w)

25w=72+2w²

2w²-25w+72=0

2*72=2*2*2*2*3*3=(2*2*2*2)(3*3) and 16+9=25

factor

(2w    )(w    )=0 so far...

(2w   9)(w  8)=0 so far...

(2w-9)(w-8)=0

w-8=0

w=8

2w-9=0

2w=9

w=9/2=4.5

in reality both answers are the answers to the problem

one is the length and one is the width

P=2(8)+2(4.5)=16+9=25

A=8*4.5=36

This is a pretty tough problem for 7th grade math.

Hope this helps. Bye.