the length of a rectangle is 3 feet less than twice its width, and its perimeter is between 24 and 48 feet

Question

the length of a rectangle is 3 feet less than twice its width, and its perimeter is between 24 and 48 feet. The answer must be expressed as an inequality. Please help!!

Answer 1

The perimeter of a rectangle is the sum of its sides.  Let's let x = the width of the rectangle.  The length of the rectangle is then 3 feet less then twice its width, or 2x - 3.

 

P = x + x + (2x - 3) + (2x - 3)

P = (1 + 1 + 2 + 2)x - 6

P = 6x - 6

P = 6(x-1)

 

We know that the perimeter of the rectangle is between 24 and 48.  We can set up the inequality:

 

24 < 6(x-1) < 48  

4 < x - 1 < 8   (Divide the entire inequality by 6)

5 < x < 9   (Add one to both sides)

Answer 2

Hello, Harry.... 

You are so lucky... Look at how many people are answering your question.  Since I solved the problem, might as well post it.

 

Let w be the width

Let l be the length

Let p be the perimeter

 

The problem states that "the length is 3 less than twice its width". This can be translated as

1) l= 2w-3

Another condition that must be met is that the perimeter must be "between" 24 and 48.  This can be expressed as:

2) 24< p< 48

 

Remember, perimeter is the distance around. So, 

3) p= l+w+l+ w= 2l+2w

 

Now that we established all the equations and inequalities that we are going to need, we are ready to go. Using equations # 2 and #3, we have

 

24< p              <  48               Substituting for p

24< 2l+ 2w      < 48                Dividing by 2

12< l+w           <24                 Substituting 2w-3 for l

12< (2w-3)+ w < 24                Combining like terms

12< 3w   -3       <24                Adding 3

+3          +3        +3

15<  3w             <27               Dividing by 3

5< w< 9                                  The width must be between 5 and 9.

 

Next, we can find the length by using equation #1. l= 2w-3.

1) l= 2w-3              Adding 3 to both sides of the equation.

    +3     +3

4) l+3= 2w.  So, since p= 2l+2w, by substitution, we get 

5) p= 2l+ l+ 3

Using equation # 2 and # 5, we get

24< 2l+l+3< 48       Combine like terms

24< 3l+3< 48          Divide by 3

8< l + 1  < 16         Subtract 1 

-1       -1    -1

7<l < 15                 The length must be between 7 and 15.

 

I know is kind of long... Hey, try to type it on an ipad like I just did. It took me forever!! But, I am here to help.  

 

D.Y. T.

dont forget to rate me :)

Answer 3

Let L = the length of the rectangle and W = its width.

 

L = 2W - 3                   The length of a rectangle is 3 feet less than twice its width

 

24 ≤ 2W + 2L ≤ 48       Its perimeter is between 24 and 48 feet (I assume "between" includes 24 and 48)

 

Where 2L + 2W is the perimeter of a rectangle.  Let's substitute 2W-3 in place of L in the second inequality:

 

24 ≤ 2W + 2(2W-3) ≤ 48

 

24 ≤ 2W + 4W - 6 ≤ 48

 

24 ≤ 6W - 6 ≤ 48

 

30 ≤ 6W ≤ 54

 

5 ft ≤ W ≤ 9 ft

 

L = 2W - 3, where W ranges from 5 to 9 feet:

 

2(5)-3 ≤ L ≤ 2(9)-3

 

7 ft ≤ L ≤ 15 ft

Answer 4

Let   L = the length and W = width.    Then  L  = 2 W -3

The perimeter, P,   is     P =  2 L + 2W  .    Substituting for L  we get

 P = 4W - 6 + 2 W  =   6 W - 6

 

Since P is between  24 and 48  ,  we can write

 

24 < 6 W - 6  < 48   .

 

This can be simplified.     Add 6 to everything to get  

 

30 < 6 W < 54         then divide through by 6 to get

 

  5 <  W  <  9

 

 

This is one form of the answer.    It can be manipulated to show that

 

 7 <  L    < 15