Question : The area (in m2) of a circular path of uniform width $x$ metres surrounding a circular region of diameter d metres is _____.
Option 1: $\pi x(x+2 \mathrm{~d})$
Option 2: $\pi x(x+\mathrm{d})$
Option 3: $\pi x(2 x+\mathrm{d})$
Option 4: $\pi x\left(x+\frac{\mathrm{d}}{2}\right)$
Correct Answer: $\pi x(x+\mathrm{d})$
Solution :
Radius of circular region ($r$) = $\frac{d}{2}$
Width of path = $x$ m
External radius of path ($R$) = $\frac{d}{2} + x$
As we know,
Area of path = $π(R^2 - r^2)$
= $\pi[(\frac{d}{2} + x)^2 - (\frac{d}{2})^2]$
= $\pi [\frac{d^2}{4} + x^2+ dx - \frac{d^2}{4}]$
= $\pi [x^2 + dx]$
= $\pi x (x + d)$ m
2
Hence, the correct answer is $\pi x (x + d)$.
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