Question

The perimeter of a rectangle is 52 feet. Describe the possible lengths of a side if the area of the rectangle is not to exceed 105 square feet.

Express the possible lenghs of a side

Answer 1

Let l is length of the rectangle and b is breadth of the rectangle.

Perimeter of the Rectangle  = 52ft

Formula for rectangle perimeter = 2 (l + b)

2(l + b) = 52

l + b = 52/2

l + b = 26

l = 26 - b ---> (1)

Area of the rectangle  = 105 ft²

Formula for rectangle area = lb

lb = 105 ---> (2)

From equation (1), substitute the value of l in equation (2).

(26 - b)b = 105

26b - b² = 105

b² - 26b + 105 = 0

Solve the above equation for b.

b² - 21b - 5b + 105 = 0

b(b - 21) -5(b - 21)  =  0

(b - 5) (b - 21)  =  0

(b - 5)  =  0    :     (b - 21)  =  0

b  =  5   :     b  =  21

Substitute b = 5 in l = 26 - b

l = 21 .

Substitute b = 21 in l = 26 - b

l = 5 .

In rectangle length is always greater than breadth.

Hence, l = 21ft and b = 5ft.

Answer :

The Dimensions of a Rectangle are l = 21ft and b = 5ft.