question_answer

A rhombus shaped sheet with perimeter 40 cm and one diagonal 12 cm, is painted on both sides at the rate of ` 5 per \[c{{m}^{2}}.\]Find the cost of painting.

A)  ` 880                          

B)         ` 1020

C)         ` 960                          

D)         ` 980              

Correct Answer: C

Solution :

Let ABCD be a rhombus having sides A = BC = CD = DA = \[x\,cm\] Perimeter of rhombus \[=40\,cm\]                    [Given] \[\Rightarrow \]\[x+x+x+x=40\] \[\Rightarrow \]\[4x=40\] \[\Rightarrow \]\[x=10\] In \[\Delta ABC,\]let \[a=10\,cm,\,b=12\,cm\]and \[c=10\,cm\] Now, semi-perimeter of \[\Delta \Alpha \Beta C,s=\frac{a+b+c}{2}\] \[=\left( \frac{10+10+12}{2} \right)cm\,=\frac{32}{2}\,cm\,=16\,cm\] \[\therefore \]    Area of \[\Delta ABC\] \[=\sqrt{16(16-10)(16-10)(16-12)}\,c{{m}^{2}}\] \[=\sqrt{16\times 6\times 6\times 4}\,c{{m}^{2}}=48\,c{{m}^{2}}\] \[=\sqrt{16\times 6\times 6\times 4}\,c{{m}^{2}}=48\,c{{m}^{2}}\] Now, area of the rhombus ABCD \[=2(Area\,of\,\Delta \Alpha \Beta C)=(2\times 48)\,c{{m}^{2}}=96\,c{{m}^{2}}\] \[\because \]Cost of painting the sheet of area \[1\,c{{m}^{2}}=\] ` 5 \[\therefore \]Cost of painting the sheet of area \[96\,c{{m}^{2}}=\]` \[(96\times 5)=\]` 480 Thus, the cost of painting the sheet on both sides = ` \[(2\times 480)=\]` 960