Question

Find the cost of laying grass in a triangular field of sides 50 m, 65m and 65m at the rate of Rs 5 per m2 .

Answer 1

⇒ Given:

A triangular field of sides 50 m, 65 m and 65 m.

The cost of laying grass in 1 m² is. Rs.5.

⇒ To Find:

The total cost of laying grass in the field.

⇒ Solution:

Side 1 [a] = 50 m

Side 2 [b] = 65 m

Side 3 [c] = 65 m

Semi-perimeter [S]:

$\sf{\implies\:Semiperimeter=\dfrac{a+b+c}{2}}$

$\sf{=\dfrac{50+65+65}{2}}$

$\sf{=\dfrac{180}{2}}$

$\sf{=90\:cm}$

Now, we need to use the Heron's formula:

$\sf{\implies\:Heron's\:Formula=\sqrt{S\times(S-a)\times(S-b)\times(S-c)}}$

Substituting the values in the equation:

$\sf{\longrightarrow\sqrt{90\times(90-50)\times(90-65)\times(90-65)}}$

$\sf{\longrightarrow\sqrt{90\times40\times25\times25}}$

$\sf{\longrightarrow25\sqrt{90\times40}}$

$\sf{\longrightarrow25\sqrt{2\times45\times2\times20}}$

$\sf{\longrightarrow50\sqrt{45\times20}}$

$\sf{\longrightarrow50\sqrt{3\times3\times5\times5\times4}}$

$\sf{\longrightarrow50\times30}$

$\sf{\longrightarrow\:Area\:of\:the\:field=1500\:m^2}$

It is given that the cost of laying grass on the field is Rs.5/m².

Total cost of laying grass:

= Area of field x Cost per m²

= 1500 x 5

= Rs.7500

∴ The total cost of laying grass on the field is Rs.7500.

Heron's Formula:

$\sf{Formula=\sqrt{S\times(S-a)\times(S-b)\times(S-c)}}$

According to the Heron's formula, we can find the area of any triangle whose three sides are given.

The meaning of the terms in the formula is as follows:

S = Semiperimeter

a = Side 1

b = Side 2

c = Side 3