Question

The area of the compound shape below is 30mm^2.

Calculate the value of x.
If your answer is a decimal, give it to 1 d.p

Answer 1

• x = 2.5 mm

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Set equation for area of given shape:

• x*4 + x*(2x + 3) = 30

Simplify and solve for x:

• 4x + 2x² + 3x = 30
• 2x²+ 7x = 30
• 2x² + 7x - 30 = 0
• x = [-7 ± √(7² - 4*2*(-30))] /4
• x = (-7 ± √289)/4
• x = (-7 ± 17)/4
• x = (10/4) and x = (- 24)/4 (discarded as negative number for the segment length)
• x = 2.5 mm

Answer 2

x = 2.5 mm

Explanation:

A compound shape is made up of two or more basic shapes.

From inspection of the given compound shape, we can see that it is made up of two rectangles.

`\boxed{\begin{minipage}{4 cm}\underline{Area of a rectangle}\\\\$A=w \cdot l$\\\\where:\\ \phantom{ww}$\bullet$ $w$ is the width. \\ \phantom{ww}$\bullet$ $l$ is the length.\\\end{minipage}}`

Dimensions of the smaller rectangle:

• w = x
• l = 4 mm

Dimensions of the larger rectangle:

• w = x
• l = (2x + 3) mm

Therefore, the equation for the area of the compound shape is:

`\begin{aligned}\implies \textsf{Area}&=x \cdot 4 + x \cdot (2x+3)\\&=4x+x(2x+3)\\&=4x+2x^2+3x\\&=2x^2+7x\end{aligned}`

Given the area of the compound shape is 30 mm², substitute this value into the found equation for area and solve for x:

`\begin{aligned}\implies 2x^2+7x &=30\\2x^2+7x-30&=0\\2x^2+12x-5x-30&=0\\2x(x+6)-5(x+6)&=0\\(2x-5)(x+6)&=0\\\\2x-5&=0 \implies x=2.5\\x+6&=0 \implies x=-6\end{aligned}`

As length is positive, x = 2.5 mm only.