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A path of 4 m width runs round a semicircular grassy plot whose circumference is $163\frac{3}{7}\ m$. Find the cost of gravelling the path at the rate of Rs. 1.50 per square metre.
Given:
A path of 4 m width runs round a semicircular grassy plot whose circumference is $163\frac{3}{7}\ m$.
To do:
We have to find the cost of gravelling the path at the rate of Rs. 1.50 per square metre.
Solution:
Width of the path around the semicircular grassy plot $= 4\ m$.
Circumference of the plot $= 81\frac{5}{7}\ m$
$=\frac{572}{7}\ m$
Let $r$ be the radius of the plot.
This implies,
$\frac{2\pi r}{2}=\frac{572}{7}$
$\Rightarrow \frac{22}{7} r=\frac{572}{7}$
$\Rightarrow r=\frac{572}{7} \times \frac{7}{22}$
$\Rightarrow r=26$
The radius of the plot is $26 \mathrm{~m}$.
Width of the path $=4 \mathrm{~m}$
Outer radius $R=26+4=30 \mathrm{~m}$
Cost of gravelling at the rate of Rs. 1.50 per $\mathrm{m}^{2}=Rs.\ 352 \times \frac{150}{100}$
$=Rs.\ \frac{352 \times 3}{2}$
$=Rs.\ 176 \times 3$
$=Rs.\ 528$
The cost of gravelling the path at the rate of Rs. 1.50 per square metre is Rs. 528.