Inequalities

Question

The perimeter of a rectangle is 220 feet. Describe the possible lengths of a side if the area of the rectangle is not to exceed 1425 square feet.

Answer 1

Let:

    L = length of rectangle

    W = width of rectangle

 

Translate:

  "perimeter of a rectangle is 220 feet"     means

         2L + 2W              = 220                   [note: P = 2L+2W]

 

  "area of the rectangle is not Let:
    L = length of rectangle
    W = width of rectangle
 
Translate:
  "perimeter of a rectangle is 220 feet"     means
         2L + 2W              = 220                   [note: P = 2L+2W]
 
  "area of the rectangle is not to exceed 1425 square feet"    means
      LW                                   ≤           1425                [note: A=LW]
             {note: "not to exceed" is "not greater than"; that is, "less than or equal to"]
 
Re-write:
    2L + 2W = 220
     L + W = 110                   [divide both sides by 2]
     W = 110 - L              [solve for W so we can get rid of it]
 
    LW ≤ 1425
    L(110-L) ≤ 1425
     110L - L² ≤ 1425
     L² - 110L ≥ -1425         [multiply by (-1) reverses sense of inequality]
     L² - 110L + 1425 ≥ 0       [add +1425 to both sides retains sense of inequality]
    (L - 15)(L - 95) ≥ 0           [either factor or use quadratic formula]
         [note: this is true when terms are ++ or -- or 00; that is, L≥95 or L≤15]
So,
  The length must be greater than or equal to 95 or less than or equal to 15.
   Since the perimeter is fixed at 220, if the length is greater than or equal to 95, then it must also be less than 110 (for there to be any width at all).
   Since the perimeter is fixed at 220, if the length is less than or equal to 15, then it must also be greater than 0.
 
The expression is:     0< L ≤15  OR  95 ≤ L < 110

 

Answer 2

Hi Kate,

If x and y are the length and width of the rectangle respectively, then; X > 0, y > 0

2x + 2y ≤ 220

  x +  y = 110 ==> y = 110 -x

       xy ≤ 1425

x(110 -x) = 1425

 110x -x² = 1425

x² -110x +1425 = 0

(x -15)(x -95) = 0

(x -15) = 0 or (x -95) = 0

x = 15 ft, x = 95ft

 

Since x is the length (the longer side) and y = 110 -x, 

x = 95, y =15

   0 ≤ y ≤ 15ft  and  95 ≤ x ≤110 ft

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