Question

A rhombus shaped field has green grass for 18 cows to graze. If each side of the rhombus is 30m and its longer diagonal is 48cm, how much area of grass field will each cow be getting? ​

Answer 1

${\bold{\huge{\underline{Question}}}}$

A rhombus-shaped field has green grass for 18 cows to graze. If each side of the rhombus is 30m and its longer diagonal is 48cm, how much area of grass field will each cow be getting?

${\bold{\huge{\underline{Solution-}}}}$

Given:-

A rhombus shaped field has green grass for 18 cows

One side of the rhombus = 30 m

Diagonal = 48 cm

In Δ ABC

a = 48cm

b = 30cm

c = 30cm

As the semi-perimeter is the half of the sum of sides of the triangle.

s = $\frac{a + b + c}{2}$

s = $\frac{48 + 30 + 30}{2}$

s = $\frac{108}{2}$

s = 54m.

Therefore area of triangle =

$= \: \sqrt{s(s - a)(s - b)(s - c)} \\ \\ = \: \sqrt{54(54 - 48)(54 - 30)(54 - 30} \\ \\ = \: \sqrt{54 \times 6 \times 24 \times 24} \\ \\ = \: \sqrt{3 \times 3 \times 6 \times 6 \times 24 \times 24} \\ \\ = \: 3 \times 6 \times {24}^{2} \\ \\ = \: 18 \times {24}^{2} \\ \\ = \: {432m}^{2}$

Therefore area of rhombus = $2 \times 432$ = ${864}^{2}$

Therefore are of the grass for 18 cows = ${864}^{2}$

Therefore area of the grass for one cow = $\frac{864}{18}$

Answer = ${48m}^{2}$

Answer 2

$\bold{\underline{\underline{Answer:}}}$

Area available for each cow for gazing in the rhombus shaped field is 48 m²

$\bold{\underline{\underline{Stpe\:-\:by\:-\:step\:explanation:}}}$

Given :-

• A rhombus shaped field has green grass for 18 cows
• One side of the rhombus = 30 m
• Diagonal = 48 cm

To find :-

• Area of grass field available for each cow

Solution :-

From the question we infer that the field is in rhombus shape.

•°• Each side of the field will be equal in length.

PS = 30 m

SR = 30 m

RQ = 30 m

QP = 30 m

Diagonal = PR = 48 m

We know the property of a rhombus, diagonals bisects each other at right angles.

$\bold{Area\:of\:rhombus\:=\:Area\:of\:\triangle\:PSQ\:+\:Area\:of\:\triangle\:QRS}$

Let's initially calculate the area of triangle PQS using heron's formula.

PS = 30 m = a

PQ = 30 m = b

QS = 48 m = c

Semiperimeter (s) = $\bold{\frac{a\:+\:b\:+\:c}{2}}$

Semiperimeter (s) =  $\bold{\frac{30\:+\:30\:+\:48}{2}}$

Semiperimeter (s) =   $\bold{\frac{108}{2}}$

Semiperimeter (s) = 54 m

Now, using the value of semiperimeter (s) we can calculate the area of triangle PQS,

A(Δ PQS) = $\sqrt{s(s-a) (s-b) (s-c)}$

A (Δ PQS) = $\sqrt{54(54-30) (54-30) (54-48)}$

A (Δ PQS) = $\sqrt{54(24) (24) (6)}$

A (Δ PQS) =  $\sqrt{54\times\:576\times\:6}$

A(Δ PQS) =  $\sqrt{54\times\:3456}$

A(Δ PQS) = $\sqrt{186624}$

A(Δ PQS) = 432 sq.m

•°• Area of triangle PQS = 432 m²

Similarly, A ( QRS) = 432 sq.m

[Since both the triangles have same sides]

$\bold{Area\:of\:rhombus\:PQRS\:=\:Area\:of\:\triangle\:PSQ\:+\:Area\:of\:\triangle\:QRS}$

$\bold{Area\:of\:rhombus\:PQRS\:=432\:+\:432}$

$\bold{Area\:of\:rhombus\:PQRS\:=\:864\:sq.m}$

Area of rhombus PQRS = 864 m²

Since, now we have to calculate the area available for each cow for gazing, we will divide the area of rhombus PQRS by the number of cows.

Area available for each cow,

$\implies$ $\bold{\frac{Area\:of\:rhombus}{Number\:of\:cows}}$

$\implies$ $\bold{\frac{864}{18}}$

$\implies$ $\bold{48}$

•°• Area available for each cow is 48 m²