Question

32. A rhombus-shaped sheet with perimeter 40 cm and one diagonal
12 cm, is painted on both sides at the rate of 5 per cm. Find the cost of
painting
​

Answer 1

AnswEr :

Rs.960.

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

A rhombus - shaped sheet with perimeter 40 cm and one diagonal 12 cm is painted on both sides at the rate of Rs.5/cm.

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

The cost of painting.

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

We know that all sides of rhombus are equal length.

Let the ABCD be a rhombus each side be r

$\bf{We\:have}\begin{cases}\sf{Perimeter\:of\:rhombus=40\:cm}\\ \sf{One\:diagonal\:of\:rhombus\:(d_{1})=12\:cm}\\ \sf{Rate\:=\:Rs.5/cm}\end{cases}$

A/q

$\bf{\large{\purple{Perimeter\:of\:rhombus\:=4\times side}}}}}$

$\leadsto\sf{40=4\times r}\\\\\\\leadsto\sf{r=\cancel{\dfrac{40}{4} }}\\\\\\\leadsto\sf{\red{r\:=\:10\:cm}}$

$\bf{\underline{\underline{\bf{Using\:Heron's\:Formula\::}}}}}$

$\bf{We\:get\:In\:\triangle ABC}\begin{cases}\sf{a=AB=10\:cm}\\ \sf{b=BC=10\:cm}\\ \sf{c=AC=12\:cm}\end{cases}$

$\mapsto\tt{Semi-perimeter\:(S)=\dfrac{a+b+c}{2} }\\\\\\\mapsto\tt{Semi-perimeter=\dfrac{10+10+12}{2} }\\\\\\\mapsto\tt{Sem-perimeter=\cancel{\dfrac{32}{2} }}\\\\\\\mapsto\tt{\red{Semi-perimeter=16\:cm}}$

Now,

$\Rightarrow\sf{Area\:of\:\triangle ABC=\sqrt{s(s-a)(s-b)(s-c)} }\\\\\\\Rightarrow\sf{Area\:of\:\triangle ABC=\sqrt{16(16-10)(16-10)(16-12)}} \\\\\\\Rightarrow\sf{Area\:of\:\triangle ABC=\sqrt{16(6)(6)(4) } \:cm^{2} }\\\\\\\Rightarrow\sf{Ara\:of\:\triangle ABC=\sqrt{2304} \:cm^{2}} \\\\\\\Rightarrow\sf{\red{Area\:of\:\triangle ABC=48\:cm^{2} }}$

∴Area of rhombus = 2(Area of ΔABC)  

→ Area = 2(48)cm²

→ Area = 96 cm²

$\bf{\boxed{\bf{The\:cost\:of\:painting\::}}}}}$

$\leadsto\sf{Cost\:of\:painting\:of\:sheet\:of\:1cm^{2} =Rs.5}\\\\\leadsto\sf{Cost\:of\:painting\:of\:sheet\:of\:96cm^{2}=Rs.(5\times 96)=Rs.480}$

∴ The both sides is painted = Rs.2(480) = Rs.960.

Answer 2

$\Huge{\underline{\underline{\mathfrak{ Solution \colon }}}}$

Given, Perimeter of the rhombus =40 cm

All the sides of rhombus are equal hence side = 4 a

Let the diagonal AC =12 cm

$: ⟹ \text{OA } = 6 \text{cm}$

$: ⟹ \text{ \ {OB} }^{2} = { \text{ AB} }^{2} - { \text{ OA}}^{2}$

$: ⟹ \text{OB} = \sqrt{ {10 }^{2} - {6}^{2} }$

$: ⟹ \sqrt{100 - 36}$

$: ⟹ \sqrt{64}$

$: ⟹ \: 8 \text{m}$

Area of sheet

$: ⟹ \frac{1}{2} \times \text{AC} \times \text{BC}$

$: ⟹ \frac{1}{2} \times 12 \times 16$

$: ⟹96 \text{ {m}}^{2}$

Cost of painting on both side at the rate of 5 per cm

$⟹ \text{Rs}.( 5 \times 2 \times 96)$

$: ⟹\text{Rs}.960$