Dimension Problem

Question

The area of a rectangle is
35 m2
, and the length of the rectangle is
8 m
less than three times the width. Find the dimensions of the rectangle.

Answer 1

Area is Length multiplied by width.

 

We know that the total area is 35.

 

You stated that the length of the rectangle is 8 m less than three times the width.  Written as an equation, that would mean that

Length = 3W - 8   We will leave width as W.

 

Let's rewrite.  

(3w-8)(w)= 35                 We are multiplying Length times width here.

(3w² -8w) = 35                 To combine like terms by multiplying.

3w² - 8w - 35 = 0              Set the equation to zero so that you can use the quadratic equation.

(3w+ 7) (1w-5)                  Now you have 2 equations that need to be set to zero

 

 

3w+7 = 0                                        1w-5=0

     -7    -7      subtract from both            +5 +5  add to both sides to get letters on one side and numbers on the 

                     sides                                 w = 5   other

                                    

     3w= -7     (area can't be                

                     negative

                    so we are done

                  using this equation)

 

Going back to our original equation.  L= 3w-8

We now know that w = 5, so therefore we substitute.

L= 3(5)  8

      15  - 8

L= 7

 

Final Result.  Length equals 7 meters and Width equals 5 meters.

 

Answer 2

Hello, Colin

 

Let  l be the length

Let  w be the width

 

Translating into algebra "the length is 8 less than 3 times the width" is l=3w-8. Careful, it is not 8-3w.

 

Eq. #1 l= 3w-8

Eq. #2 a= l w

 

So, substituting eq. #1 into eq. #2, we get;

 

Eq. #3 35= (3w-8)w                Using the distributive property and substituting a with 35.

          35= 3w^2-8w               Subtracting 35 from both sides

          -35               -35

           0=  3w^2-8w-35          Factoring

           0= (3w+7) (w-5)        

 

  So, we have two answers:

a) w-5+= 0                                      b) 3w+7= 0

        w=5                                                 3w= -7

                                                                  w= -7/3     This cannot be an answer since dimensions cannot be         

                                                                                    negative.

 

So, we know that w=5, to find l , we used eq. #1. l= 3w-8.  l=3(5)-8=7.  The length is 7 and the width is 5.

 

I hope this helps and thanks for posting.

 

D. Y. Taylor

don't forget to rate me :)

Answer 3

Hi Colin,

 

This will not have answer as integers.  Remember, dimensions must be positive.

 

A =  L * W

 

A =  (W-8) W

 

A = W² -8W

 

35 = W² - 8W

 

0 = W2 -8W -35

 

Use quadratic formula and you will get this:  (4 + √51)

 

(4 + √51) for width & (4 + √51 -8) for length.

 

Let's check it.

 

Rewrite a little bit.

 

(√51 + 4) (√51 - 4) = difference of squares (a + b) (a - b) = a² - b²

51 -16 = 35 and it does check out!

 

There you have it.  Good luck.