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Sides of a triangular field are $ 15 \mathrm{~m}, 16 \mathrm{~m} $ and $ 17 \mathrm{~m} $. With the three corners of the field a cow, a buffalo and a horse are tied separately with ropes of length $ 7 \mathrm{~m} $ each to graze in the field. Find the area of the field which cannot be grazed by three animals.
Given:
Sides of a triangular field are \( 15 \mathrm{~m}, 16 \mathrm{~m} \) and \( 17 \mathrm{~m} \).
With the three corners of the field a cow, a buffalo and a horse are tied separately with ropes of length \( 7 \mathrm{~m} \) each to graze in the field.
To do:
We have to find the area of the field which cannot be grazed by three animals.
Solution:
Radius of each sector $r= 7\ m$
Area of sector with $\angle \mathrm{C}=\frac{\angle \mathrm{C}}{360^{\circ}} \times \pi r^{2}$
$=\frac{\angle \mathrm{C}}{360^{\circ}} \times \pi \times(7)^{2}$
Area of the sector with $\angle \mathrm{B}=\frac{\angle \mathrm{B}}{360^{\circ}} \times \pi r^{2}$
$=\frac{\angle \mathrm{B}}{360^{\circ}} \times \pi \times(7)^{2}$
Area of the sector with $\angle \mathrm{H}=\frac{\angle \mathrm{H}}{360^{\circ}} \times \pi r^{2}$
$=\frac{\angle \mathrm{H}}{360^{\circ}} \times \pi \times(7)^{2}$
Therefore,
Sum of the areas of the three sectors $=\frac{\angle \mathrm{B}}{360^{\circ}} \times \pi \times(7)^{2}+\frac{\angle \mathrm{C}}{360^{\circ}} \times \pi \times(7)^{2}+\frac{\angle \mathrm{H}}{360^{\circ}} \times \pi \times(7)^{2}$
$=\frac{(\angle \mathrm{B}+\angle \mathrm{C}+\angle \mathrm{H})}{360^{\circ}} \times \pi \times 49$
$=\frac{180^{\circ}}{360^{\circ}} \times \frac{22}{7} \times 49$
$=11 \times 77$
$=77 \mathrm{~cm}^{2}$
$a=15, b=16$ and $c=17$
$s=\frac{a+b+c}{2}$
$\Rightarrow s=\frac{15+16+17}{2}$
$=\frac{48}{2}=24$
Therefore,
Area of the triangular field $=\sqrt{s(s-a)(s-b)(s-c)}$
$=\sqrt{24 \times 9 \times 8 \times 7}$
$=\sqrt{64 \times 9 \times 21}$
$=8 \times 3 \sqrt{21}$
$=24 \sqrt{21}$
Area of the field which cannot be grazed by the three animals $=$ Area of triangular field $-$ Area of three sectors
$=24 \sqrt{21}-77 \mathrm{~m}^{2}$
The required area of the field which cannot be grazed by the three animals is \( (24 \sqrt{21}-77) \mathrm{m}^{2} \).
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