Question

the perimeter of a triangular field is 540 and its side are in the ratio 25 :17 :12 find the area of the field also find the cost of plugging the field at 40 per 100m square​

Answer 1

Solution :

$\bf{\red{\underline{\bf{Given\::}}}}$

The perimeter of a triangular field is 540 m and it's side are in the ratio 25:17:12.

$\bf{\red{\underline{\bf{To\:find\::}}}}$

The area of the field and the cost of plugging the field Rs.40/100m².

$\bf{\red{\underline{\bf{Explanation\::}}}}$

We know that formula of the perimeter of triangle :

$\boxed{\bf{Perimeter=Side+Side+Side}}}}$

Show diagram :

$\setlength{\unitlength}{1cm}\begin{picture}(6,8)\linethickness{0.075mm}\put(1,.5){\line(2,1){3}}\put(4,2){\line(-2,1){2}}\put(2,3){\line(-2,-5){1}}\put(.7, .3){ A }\put(4.05,1.9){ B }\put(1.7,2.95){ C }\put(3.2 ,2.5){ 17\:m }\put(0.6,1.7){ 12\:m }\put(2.7,1.05){ 25\:m }\end{picture}$

A/q

Let the ratio be r

$\dashrightarrow\tt{25r+17r+12r=540}\\\\\dashrightarrow\tt{54r=540}\\\\\dashrightarrow\tt{r=\cancel{\dfrac{540}{54} }}\\\\\\\dashrightarrow\tt{\pink{r=10\:m}}$

So;

$\bullet\sf{1_{st}\:side=25r=25\times 10=\boxed{\sf{250\:m}}}}\\\bullet\sf{2_{nd}\:side=17r=17\times 10=\boxed{\sf{170\:m}}}}\\\bullet\sf{3_{rd}\:side=12r=12\times 10=\boxed{\sf{120\:m}}}}$

__________________________________________

$\underline{\underline{\bf{\green{Using\:Heron's\:formula\::}}}}$

$\longrightarrow\tt{Semi\:Perimeter=\dfrac{Sum\:of\:side}{2}} \\\\\\\longrightarrow\tt{Semi\:Perimeter=\dfrac{a+b+c}{2} }\\\\\\\longrightarrow\tt{Semi\:Perimeter=\dfrac{250cm+170cm+120cm}{2} }\\\\\\\longrightarrow\tt{Semi\:Perimeter=\cancel{\dfrac{540}{2} }m}\\\\\\\longrightarrow\tt{\blue{Semi\:Perimeter=270m}}$

&

$\leadsto\tt{Area\:of\:\triangle=\sqrt{s(s-a)(s-b)(s-c)} }\\\\\leadsto\tt{Area\:of\:\triangle=\sqrt{270(270-250)(270-170)(250-120)}}\\ \\\leadsto\tt{Area\:of\:\triangle=\sqrt{270(20)(100)(150)} }\\\\\leadsto\tt{Area\:of\:\triangle=\sqrt{81000000} }\\\\\leadsto\tt{\blue{Area\:of\:\triangle=9000m^{2} }}$

$\underline{\underline{\bf{\green{Cost\:of\:ploughing\:the\:field\::}}}}$

$\leadsto\tt{100m^{2} =Rs.40}\\\\\leadsto\tt{1m^{2} =\dfrac{4\cancel{0}}{10\cancel{0}}} \\\\\leadsto\tt{9000m^{2} =\dfrac{4}{\cancel{10}} \times 900\cancel{0}}\\\\\leadsto\tt{Rs.(4\times 900)}\\\\\leadsto\tt{\blue{Rs.3600}}$

Thus;

The cost of ploughing field is Rs.3600 .

Answer 2

Given:

Perimeter of a triangular field = 560

Sides are in the ratio = 25:17:12

To find:

the area of the field also the cost of plugging the field at Rs. 40 per 100m²

Solution:

Let the side be 25x,17x,12x

perimeter of triangle = 25x+ 17x+12x

→540 = 25x+ 17x+ 12x

→540 = 54x

→540/54 = x

→x = 10m

Sides of triangle

→25x= 25×10= 250

→17x = 17× 10 = 170

→12x = 12× 10 = 120

Semi perimeter = 540/2= 270

by herons formula,

Area of triangle field = √[s(s-a)(s-b)(s-c)]

→√[270(270-250)(270-170)(270-250)

→√[270(20)(100)(150)]

→√270×20×100×150

→√81000000

→9000m²

hence the area of the field= 9000m²

the cost of plugging the field at 40 per 100m square= 40/100×9000

→0.4×9000=3600

Hence the cost of plugging= 3600Rs