The cost of turfing a triangular field at the rate of Rs 5 per sq meter is Rs 6720. If the sides of the field are in the ratio 4:7:9, find the sides of the field.

Answer :

Given 
Rate of turfing a triangular field = Rs 5 per square meter
​Total cost of turfing = Rs. 6720 
ration of sides = 4 : 7 : 9
So,
Area of triangular field = $\frac{Total \cos t}{Rate}$  = $\frac{6720}{5}$  = 1344 m^​2

Let or side according to ratio = 4 x , 7 x And 9x
Now we use herons's formula 
Area of triangle = $\sqrt{s \left(s - a\right)\left(s - b\right)\left(s - c\right)}$
Where
s = $\frac{a + b + c}{2}$
And
a  = 4x  , b = 7x And c = 9x 
So,
s = $\frac{4x + 7x + 9x}{2}$  = 10x 

Area of triangular field = $\sqrt{10x \left(10x - 4x\right)\left(10x - 7x\right)\left(10x - 9x\right)}$
 
$\Rightarrow$ $\sqrt{10x \left(10x - 4x\right)\left(10x - 7x\right)\left(10x - 9x\right)}$ = 1344

$\Rightarrow$$\sqrt{180x^4}$ = 1344

Taking whole square on both hand side , we get

$\Rightarrow$180x⁴ = ( 1344 )²

$\Rightarrow$180x⁴ = 1806336

$\Rightarrow$x⁴ = 10035.2

$\Rightarrow$x = 10.008

$\Rightarrow$ x = 10

So, 
Sides of triangular field are = 40 m , 70 m And  90 m                       (Ans)