A path of 4 m width runs round a semi­circular grassy plot whose circumference is $163\frac{3}{7}\ m$. Find the cost of turfing the plot at the rate of 45 paise per $m^2$.

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 turfing the plot at the rate of 45 paise per $m^2$.

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 $R=26+4=30 \mathrm{~m}$

Area of the plot $=\frac{1}{2} \pi r^{2}$

$=\frac{1}{2} \times \frac{22}{7} \times(26)^{2} \mathrm{~m}^{2}$

$=\frac{11}{7} \times 676$

$=\frac{7436}{7} \mathrm{~m}^{2}$
Rate of turfing the plot $=45$ paisa per $\mathrm{m}^{2}$

Total cost of turning $=Rs.\ \frac{7436}{7} \times \frac{45}{100}$

$=Rs.\ 478.02$

$=Rs.\ 478$

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Updated on: 10-Oct-2022

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