Perimeter of a rectangle

Question

The perimeter of a rectangle is 34 feet and its area is 60 square feet. Find the lenght and the width of the rectangle.

Answer 1

The perimeter of a rectangle is the sum of twice the length and twice the width.  The area of a rectangle is length times width.

 

Let x = width

Let y = length

 

Using these variables, we can set up equations that represent the perimeter and area.

 

2x + 2y = 34

 

2(x + y) = 34        

 

x + y = 17        --->   perimeter

 

 

xy = 60     --->   area

 

 

We have two equations to work with.

 

 

x + y = 17              eq1

xy = 60                   eq2

 

 

Using the substitution method, we can substitute eq1 into eq2.  Lets get eq2 in terms of x.

 

x(17 - x) = 60

 

-x² + 17x = 60

 

 

Subtract 60 on both sides of the equation.

 

-x² + 17x - 60 = 0

 

 

Divide both sides of the equation by -1.  We do this so that we can have a positive x² term.  This makes it easier to use FOIL when factoring.

 

x² - 17x + 60 = 0

 

 

Now we are in the position to factor use FOIL.

 

(x - 12)(x - 5) = 0

 

 

x = 12          and         x = 5

 

 

As you can see, we have two positive x values.  These values are acceptable.  So we will have two values for width.

 

width = 5                      width = 12

 

 

Substituting the value of x into eq1, we get

 

y = 17 - 5                       y = 17 - 12

 

y = 12                            y = 5

 

 

These are the values of length.  Therefore, your combinations are

 

length = 5 feet                             length = 12 feet

width = 12 feet                            width = 5 feet

 

 

The reason for this combination is because the length can be smaller than the width, or the length can be larger than the width.  It has to do with orientation.

Answer 2

Hi Hann,

 

Let L= Length

     W = Width

     A = Area

     P = perimeter

 

Perimeter of a rectangle is the sum of the sizes of the four sides, 2W and 2L

P = 2W+2L

2W+2L = 34.....W+L= 17

W = 17-L...............................................(1)

A = L*W

L*W = 60

W = 60/L...............................................(2)

 

Using the above two expressions we have

60/L = 17-L

 

Multiply both sides by L

60 = 17L-L^2....rearranging the above expression we have

L^2-17L+60 = 0

(L-12) (L-5) = 0

 

L = 12 or L = 5

 

Depending on the value of L we take, we can determine W as

 

W = 60/L = 60/5 = 12

          or

W = 60/L = 60/12 = 5

 

Therefore, the rectangle is a 12 by 5.

 

I hope it helps

A. 

Answer 3

P=2l+2w

A=lw

34=2l+2w

60=lw

34=2l+2w

17=l+w

17-l=w

60=lw

substitute

60=l(17-l)

60=17l-l^2

l^2-17l+60=0

60=2*2*3*5=(2*2*3)*(5)=12*5 and 12+5=17, the coefficient of the middle term

factor

(l-12)(l-5)=0

l-12=0

l=12 feet

l-5=0

l=5 feet

the length and width are:

length=12 feet

width=5 feet

34=2*12+2*5=24+10=34

60=5*12=60

this is all of the work (algebra) needed to solve the problem

however, you could have solved the problem using simple arithmetic

if the perimeter is 34, then half the perimeter is 17

find 2 numbers that add up to 17 and multiply to make 60

use trial and error or factor 60 like I did before

60=2*2*3*5=(2*2*3)*(5)=12*5=60 and 12+5=17