A rhombus shaped sheet with perimeter `40` `cm` and one diagonal `12` `cm`, is painted on both sides at the rate of `Rs` `5` per `cm^(2)`. Find the cost of painting.

Explanation

Let ABCD be a rhombus having each side equal to x cm.
`"i.e.," " "AB=BC=CD=DA= x cm.`
`"Given, perimeter of a rhombus = 40" `

`therefore" "AB+BC+CD+DA=40`
`rArr" "x+x+x+x+=40`
`rArr" "4x=40`
`rArr" "x=(40)/(4)`
`therefore" "x=10cm`
`"In " triangleABC," let "" a=AB=10cm,b=BC=10cm"`
`"and"" "c=AC=10cm,b=BC=10cm`
`"Now,""semi-perimeter of a "triangleABC,s=(a+b+c)/(2)`
`=(10+10+12)/(2)=(32)/(2)=16cm`
`therefore" ""Area of "triangleABC=sqrt(s(-a)(s-b)(s-c))" "["by Heron's formula"]`
`=sqrt(16(16-10)(16-10)(16-12))`
`=sqrt(16xx6xx6xx4=4xx6xx2=48cm^(2))`
`therefore" ""Area of the rhombus"=2("Area of "triangleABC)=2xx48`
`=96cm^(2)`
`because" ""Cost of painting of the sheet of " 1 cm^(2)="rupee "5`
`therefore" ""Cost of painting of the sheet of " 96 cm^(2)=96xx5="rupee " 480`
`"Hence, the cost of the painting of the sheet for both sides" =2xx480= "rupee " 960`