Question

3. A rectangular garden has an area of 40m² and a perimeter of 26m. The length of the

garden is a cm and its width is b cm.
(a) Form two equations involving a and b.
(b) Solve these equations simultaneously in order to find the length and width
of the garden

Answer 1

Explanation:

Perimeter = 2length + 2 width

2a + 2b = 26 m

Divide the equation by 2

`(2a)/(2)+(2b)/(2)=(26)/(2)\\\\`

a + b = 13 --------------(I)

a = 13 -b

Area = length *width

a*b = 40 sq.m

ab = 40 ----------------(II)

Plugin a = 13 -b in the equation (II)

(13 - b)*b = 40

13*b - b*b = 40

13b - b² = 40

0 = 40 + b² - 13b

b² - 13b + 40 = 0

Sum = - 13

Product = 40

Factors = (-8) , (-5) {-8 + (-5) = -13 & (-5)*(-8) = 40}

b² - 8b - 5b + (-8)*(-5) = 40

b(b - 8) - 5(b - 8)= 0

( b - 8) (b - 5) = 0

b - 8 =0 ; b -5 =0

b= 8 ; b = 5

Length = 8 cm

Width = 5cm

Answer 2

a) a +b= 13, ab= 40

b) length= 8m, width= 5m

Explanation:

Perimeter of rectangle= 2(length +width)

Area of rectangle= length ×width

a) Perimeter= 26m

2(a +b)= 26

Divide both sides by 2:

a +b= 26 ÷2

a +b= 13 ----- (1)

Area= 40m²

ab= 40 ----- (2)

b) From (1): a= 13 -b ----- (3)

Substitute (3) into (2):

(13 -b)(b)= 40

Expand:

13b -b²= 40

b² -13b +40= 0

Factorise:

(b -5)(b -8)= 0

b -5= 0 or b -8= 0

b= 5 or b= 8

Substitute into (3):

a= 13 -5 or a= 13 -8

a= 8 or a= 5

Since the length is usually the longer side of the rectangle, the length and the width of the rectangular garden is 8m and 5m respectively.