Question

Three cows are tethered with 10 m long rope at the three corners of a triangular field

having sides 42 , 20 m and 34 m. Find the anta of the plot which can be grazed by the

cows also find the area of the remaining field (ungrazed).​

Answer 1

Step-by-step explanation:

$\mathcal{Suppose \: ABC \: is \: the \: triangular \: field.}$

$\sf{\color{black}{Let \: AB \: = 42m, \: AC = 20m \: and \: BC = 34m.}}$

$\mathcal {Length \: of \: the \: rope, \: r = 10 m}$

We know that , ◉‿◉

${\color{blue}\boxed{\red {\mathcal{Area \: of \: sector=\frac{\theta}{360} \times πr^2}}}}$

So,

Area of the field grazed by cow tethered at A $\bf=\frac{A}{360} \times πr^2$

Area of the field grazed by cow tethered at B$\bf=\frac{B}{360} \times πr^2$

Area of the field grazed by cow tethered at C$\bf=\frac{C}{360} \times πr^2$

∴ Area of the plot grazed by the cow

$\sf\orange{ = \frac{A}{360} \times πr^2 + \sf\frac{B}{360} \times πr^2 + \sf\frac{C}{360} \times πr^2}$

$\sf\orange{ = \frac{(A+B+C)}{360} \times πr^2}$

$\sf\purple{ = \frac{180}{360} \times \frac{22}{7} \times (10)^2} \: \: \: (∵A + B + C= 180)$

$\sf\purple{ = \frac{\cancel{180}}{\cancel{360}} \times \frac{\cancel{22}}{7} \times ({100})}\bf \: \: \implies157.142$

$\bf{\therefore The \: area \: of \: the \: plot \: grazed \: by \: the \: cow \: is \: approx \: 157.14m^2}$

Answer 2

★Given:-

• Three cows are tethered with along rope at the three corners of a triangular field.
• Length of rope = 10m
• Sides of triangular field = 42,20&34m

★To find:-

• The area of the plot which can be grazed by the  cows.
• The area of the remaining field (ungrazed).

★Solution:-

Let ABC is  the triangular field.Then,  

• AB = 20m
• AC = 34m
• BC = 42m
• Length of the rope = radius = r

Using the formula,

✦Area of sector = θ/360° ×πr²

Area of field grazed by cow tethered at A:-

⇒A/360° ×πr²

Area of field grazed by cow tethered at B:-

⇒B/360° ×πr²

Area of field grazed by cow tethered at C:-

⇒C/360° ×πr²

The total area of field grazed by cow = Area of three sectors

⇒(A/360° ×πr² )+(B/360° ×πr²)+(C/360° ×πr²)

Taking out (1/360° ×πr²) as common,

⇒ (A+B+C)/360° ×πr²

✦Sum of all angles of a triangle is 180°

∴[A+B+C=180°]

⇒180°/360° × πr²

⇒ 180°/360° × (22/7) ×10²

⇒ 157.14m²

Total area of triangle:-

Using heron's formula,

✦Ar.Δ = √s(s-a)(s-b)(s-b)

Where, s=(a+b+c)/2

Putting values,

⇒ s = (a+b+c)/2

⇒ s = (42+20+34) /2

⇒ s = 48

Now,

⇒Area = √s(s-a)(s-b)(s-b)

⇒√48(48-42)(48-20)(48-34)

⇒√48(6)(28)(14)

⇒ 336m²

Area of field ungrazed = Area of field - Area of field grazed

⇒ 336 - 157.14

⇒ 178.86 m²

Hence,

Area of plot grazed by cows is 157.14m²

Area of remaining field is 178.96m²

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