Question

Sides of a right triangular field are 25m, 24m and 7m. At the three corners of the field, a cow, a buffalo and a horse are tied separately with ropes of 3.5 m each to graze in the field. Find the area of the field that cannot be grazed by these animals

Answer 1

AnswEr :

⋆ Sides of Right Triangular Field :

• Hypotenuse = 25 m
• Perpendicular = 24 m
• Base = 7 m

◗ A Cow, A Buffalo, and A Horse are tied separately with ropes of 3.5 m at three corners.

⋆ Refrence of Image is in the Diagram :

$\setlength{\unitlength}{1.5cm}\begin{picture}(6,2)\put(7.7,2.9){\large{A}}\put(7.7,1){\large{B}}\put(10.6,1){\large{C}}\put(8,1){\line(1,0){2.5}}\put(8,1){\line(0,2){1.9}}\put(10.5,1){\line(-4,3){2.5}}\put(7.3,2){\mathsf{\large{24 m}}}\put(9,0.7){\matsf{\large{7 m}}}\put(9.4,1.9){\mathsf{\large{25 m}}}\put(8.2,1){\line(0,1){0.2}}\put(8,1.2){\line(3,0){0.2}}\end{picture}$

$\rule{150}{1}$

☯$\:\:\underline{\textsf{Area of Right Triangular Field ABC :}}$

$\dashrightarrow\:\:\tt Area_{\tiny ABC} =\dfrac{1}{2}\times Base \times Height \\\\\\\dashrightarrow\:\: \tt Area_{\tiny ABC} =\dfrac{1}{2} \times7 \times 24 \\\\\\\dashrightarrow\:\: \tt Area_{\tiny ABC} =\dfrac{1}{2} \times168 \\\\ \\\dashrightarrow\:\:\tt Area_{\tiny ABC} = 84 \: {m}^{2}$

⠀

$\therefore\:\underline{\textsf{Total Area of Grass Field is \textbf{84 m ^2 .}}}$

$\rule{200}{1}$

$\text{Now, if these animals are tied at the}\\\text{corners and will make Sector i.e.}\\ \text{(3 Sectors of Radius 3.5 m), So will Find} \\ \text{the Area that Animals can actually Graze.}$

☯$\:\:\underline{\textsf{Area that Animals can Graze are :}}$

$:\implies\tt Area = \dfrac{\angle A}{360\degree}\pi {r}^{2} + \dfrac{\angle B}{360\degree}\pi {r}^{2} + \dfrac{\angle C}{360\degree}\pi {r}^{2}\\\\\\:\implies\tt Area = \pi {r}^{2} \bigg(\dfrac{\angle A}{360\degree}+ \dfrac{\angle B}{360\degree} + \dfrac{\angle C}{360\degree} \bigg)\\\\\\:\implies\tt Area = \pi {r}^{2} \bigg(\dfrac{\angle A +\angle B +\angle C}{360\degree}\bigg)\\\\\\:\implies\tt Area = \pi {r}^{2} \bigg(\dfrac{180 \degree}{360\degree}\bigg)\\\\\\:\implies\tt Area = \dfrac{22}{7} \times 3.5 \times 3.5 \times \dfrac{1}{2}\\\\\\:\implies\tt Area =19.25 \: {m}^{2}$

⠀

$\therefore\:\underline{\textsf{Total Area that can be Grazed is \textbf{19.25 m ^2 .}}}$

$\rule{220}{2}$

☢$\:\:\underline{\textsf{AREA THAT CAN'T BE GRAZED IS :}}$

$\twoheadrightarrow\:{\footnotesize\texttt{Area Can't Grazed = Total Area of Field - Area Can Graze}}\\\\\\\twoheadrightarrow \:\tt Area\: Can't\:Grazed = (84-19.25) \:m^2\\\\\\\twoheadrightarrow \: \underline{\boxed{\red{\tt Area\:Can't\:Grazed =64.75 \:m^2}}}$

⠀

$\therefore\:\underline{\textsf{Area that cannot be Grazed is \textbf{64.75 m ^2 .}}}$

Answer 2

Answer:

★ Area of triangle = 24*7/ 2 = 84

Now Total Angle is 180° (Triangle)

• So Grazed Area :

⇒ πr² /2

⇒ 22/7 * 3.5 * 3.5 * 1/2

⇒ 11*3.5/2

⇒ 38.5/2

⇒ 19.25

• So Area Not Grazed :

⇒ 84 – 19.25

⇒ 64.75 m²