3 cows are tethered with 10m long rope at the 3 corners of a triangular field having sides 42m, 34m, & 20m. Find the area of the plot which can be grazed by the cows.
Answers
The area of the plot which can be grazed by the cows is 157.14 m².
Step-by-step explanation:
The 3 sides of the triangular field are given as 42m, 34 m & 20 m
The length of the rope = 10m
Since the 3 cows are tethered with 10 m long rope at each vertex of the triangular field so, we have to find the area of the sector at each vertex with a radius, r = 10 m.
Let’s assume the angles of the triangle at each vertex be denoted as “θ1”, “θ2” & “θ3”.
Therefore,
The area of the field that can be grazed by cows is given by,
= Area of the 3 sectors
= [(θ1/360)×πr²] + [(θ2/360)×πr²] + [(θ3/360)×πr²]
= [πr²/360] × [θ1 + θ2 + θ3]
Since sum of angles of a triangle is 180°, ∴ θ1 + θ2 + θ3 = 180°
= [{(22/7)*10²}/360] [180°]
= [(22/7)*10²] * 0.5
= 157.14 m²
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In each corner of triangular field a horse is tethered with a rope of 7 m length. Sides of triangle are 60m ,80m ,100m.find area of field iver whuch horse can graze ?
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Three horses are tethered with 7 meter long ropes at three corners of triangular field having sides 20 meter, 34 meter and 42 meter. find the area of the plot which can be grazed by horses. also find the area of the plot which remain ungrazed.
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★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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