Question

Question 9
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.

Answer 1

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
$\Rightarrow$ x + x + x + x = 40
$\Rightarrow$ 4x = 40
$\Rightarrow x=\frac{40}{4}$
$\therefore x=10cm$

In $\Delta ABC$,
let a = AB = 10 cm, b = BC = 10 cm and c = AC = 12 cm

Now, semi-perimeter of
$\Delta ABC,s=\frac{a+b+c}{2}$
$=\frac{10+10+12}{2}=\frac{32}{2}=16cm$
$\therefore Areaof\Delta ABC=\sqrt{s(s-a)(s-b)(s-c)}byHero{n}^{\prime }sformula$
$=\sqrt{16(16-10)(16-10)(16-12)}$
$=\sqrt{16\times 6\times 6\times 4}=4\times 6\times 2=48c{m}^2$
$\therefore$ Area of the rhombus = $2\left(Areaof\Delta ABC\right)=2\times 48$
$=96c{m}^2$
The cost of painting a sheet of $1c{m}^2$ = ₹ 5
$\therefore$ The cost of painting a sheet of $96c{m}^2=96\times 5=₹480$
Hence, the cost of painting the sheet on both sides = 2 $\times$ 480 = ₹ 960