Question

The length of the rectangle below is
(3x+5)
(3x+5)

cm and its width is
( 2x+2)
2x+2

cm.







(i) Write an expression for the area of the rectangle in the form
ax2+bx+c

(ii) Given that the area of the rectangle is 32 cm2, determine the value of
x

(iii) Hence, state the dimensions of the rectangle, in centimeters.

Answer 1

i) By applying the formula for area of a rectangle, the expression for the area of the rectangle is: $\mathbf{Area = 6x^2+16x+10}$

ii) By factorization, the value of x is the area of the rectangle is 32 sq. cm is: x = 1

iii) Dimensions of the rectangle are:

• Length = 8 cm
• Width = 4 cm

Recall:

• Area of rectangle = length x width

Given the dimensions of a rectangle:

• Length = (3x+5) cm
• Width = (2x+2) cm

i) Substitute to find the expression for the area of the rectangle as $ax^2+bx+c$

• Thus:

$Area = (3x + 5)(2x + 2)$

• Expand

$\mathbf{Area = 6x^2+16x+10}$

ii) If area is given as 32 sq. cm, to find x, plug in 32 for area in $\mathbf{Area = 6x^2+16x+10}$

• Thus:

$32 = 6x^2+16x+10\\\\6x^2+16x+10 = 32\\\\6x^2+16x+10 - 32 = 0\\\\6x^2+16x - 22 = 0$

• Factorize

$2(3x^2+8x-11)\\\\2(3x+11)(x-1)$

Using (x - 1) as a factor, we can find the value of x

• Thus:

x - 1 = 0

x = 1

iii) Find the dimensions of the rectangle by plugging in the value of x:

• Length = (3x+5) = 3(1) + 5 = 8 cm
• Width = (2x+2) = 2(1) + 2 = 4 cm

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