Question

(i) A path of 8 m width runs around the outside of a circular park whose radius is 17 m. Find the area of the path.
(ii) A park is of the shape of a circle of diameter 7 m. It is surrounded by a path of width of 0.7 m. Find the expenditure of cementing the path, if its cost is ₹ 110 per sq m.

Answer 1

(i) The radius (r) of the inner circle is 17 m.
The radius (R) of the outer circle is 25 m. [Includes path, i.e., (17 + 8)]

Area of the path = $\pi R^2-\pi r^2$
$$\begin{gathered} =\mathrm{\pi }\left(R^2-r^2\right) \\ =\frac{22}{7}\left({25}^2-{17}^2\right) \\ =\frac{22}{7}\times (25-17)(25+17) \\ =\frac{22}{7}\times 8\times 42 \\ =1056{\mathrm{m}}^2 \end{gathered}$$

∴ Area of the path = 1056 m²

(ii)


Diameter of the circular park = 7 m
∴ Radius of the circular park, r = $\frac{7}{2}$ = 3.5 m
Width of the path = 0.7 m
∴ Radius of the park including the path, R = 3.5 + 0.7 = 4.2 m
Area of the path
$$\begin{gathered} =\mathrm{\pi }{R}^2-\mathrm{\pi }{r}^2 \\ =\mathrm{\pi }\left(R^2-r^2\right) \end{gathered}$$
$$\begin{gathered} =\frac{22}{7}\left(4.2^2-3.5^2\right) \\ =\frac{22}{7}\times \left(4.2-3.5\right)\times \left(4.2+3.5\right) \\ =\frac{22}{7}\times 0.7\times 7.7 \\ =16.94{\mathrm{m}}^2 \end{gathered}$$
Rate of cementing the path = Rs 110/m² (Given)
∴ Total cost of cementing the path
= 16.94 × 110
= Rs 1863.40
Thus, the expenditure of cementing the path is Rs 1863.40.