Question
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 $m^2$. Find the cost of painting.
✓
Answer
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 $= 40 \therefore\text{ AB}+\text{BC}+\text{CD}+\text{DA}=40$
$\Rightarrow\ \text{x}+\text{x}+\text{x}+\text{x}=40$
$\Rightarrow4\text{x}=40$
$\Rightarrow\text{x}=\frac{40}{4}$
$\therefore\ \text{x}=10\text{cm}$ In $\triangle\text{ABC},$
let $\text{a}=\text{AB}=10\text{cm},\text{ b}=\text{BC}=10\text{cm}$ and $\text{c}=\text{AC}=12\text{cm}$
Now, semi-perimeter of a
$\triangle\text{ABC},\text{ s}=\frac{\text{a}+\text{b}+\text{c}}{2}$
$=\frac{10+10+12}{2}=\frac{32}{2}=16\text{cm}$
$\therefore\text{ Area of }\triangle\text{ABC}=\sqrt{\text{s}(\text{s}-\text{a})(\text{s}-\text{b})(\text{s}-\text{c})}$ [by Heron's formula]
$=\sqrt{16(16-10)(16-10)(16-12)}$
$=\sqrt{16\times6\times6\times4}$
$=4\times6\times2=48\text{cm}^2$
$\because$ Cost of painting of the sheet of $1cm^2 = Rs. 5cm$
$\therefore$ Cost of painting of the sheet of $96cm^2 = 96 \times 5 = Rs. 480$
Hence, the cost of the painting of the sheet for both sides
$= 2 \times 480 = Rs. 960$
i.e., $AB = BC = CD = DA = x$ cm
Given, perimeter of a rhombus $= 40 \therefore\text{ AB}+\text{BC}+\text{CD}+\text{DA}=40$
$\Rightarrow\ \text{x}+\text{x}+\text{x}+\text{x}=40$
$\Rightarrow4\text{x}=40$
$\Rightarrow\text{x}=\frac{40}{4}$
$\therefore\ \text{x}=10\text{cm}$ In $\triangle\text{ABC},$
let $\text{a}=\text{AB}=10\text{cm},\text{ b}=\text{BC}=10\text{cm}$ and $\text{c}=\text{AC}=12\text{cm}$
Now, semi-perimeter of a
$\triangle\text{ABC},\text{ s}=\frac{\text{a}+\text{b}+\text{c}}{2}$
$=\frac{10+10+12}{2}=\frac{32}{2}=16\text{cm}$
$\therefore\text{ Area of }\triangle\text{ABC}=\sqrt{\text{s}(\text{s}-\text{a})(\text{s}-\text{b})(\text{s}-\text{c})}$ [by Heron's formula]
$=\sqrt{16(16-10)(16-10)(16-12)}$
$=\sqrt{16\times6\times6\times4}$
$=4\times6\times2=48\text{cm}^2$
$\because$ Cost of painting of the sheet of $1cm^2 = Rs. 5cm$
$\therefore$ Cost of painting of the sheet of $96cm^2 = 96 \times 5 = Rs. 480$
Hence, the cost of the painting of the sheet for both sides
$= 2 \times 480 = Rs. 960$
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