Question

The edges of a triangular board are 6cm ,8cm and 10cm. the cost of painting it at the rate of 9 paise per cm square is ___​

Answer 1

Given :

• 1st edge of the Triangle = 6 cm
• 2nd edge of the Triangle = 8 cm
• 3rd edge of the Triangle = 10 cm
• Rate of painting = 9 paise cm²

To Find :

• Cost of Painting = ?

SolutioN :

~ Formula Used :

• ${\underline{\boxed{\pmb{\sf{ S = \dfrac{a + b + c}{2} }}}}}$

• ${\underline{\boxed{\pmb{\sf{ Area = \sqrt{s (s - a)(s - b)(s - c)} }}}}}$

Where :

• s = Semi - Perimeter
• a = Side 1
• b = Side 2
• c = Side 3

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~ Calculating the Semi - Perimeter :

${\implies{\qquad{\sf{ S = \dfrac{a + b + c}{2} }}}} \\ \\ \\ \ {\implies{\qquad{\sf{ S = \dfrac{6 + 8 + 10}{2} }}}} \\ \\ \\ \ {\implies{\qquad{\sf{ S = \dfrac{6 + 18}{2} }}}} \\ \\ \\ \ {\implies{\qquad{\sf{ S = \dfrac{24}{2} }}}} \\ \\ \\ \ {\implies{\qquad{\sf{ S = \cancel\dfrac{24}{2} }}}} \\ \\ \\ \ {\qquad \; \; {\therefore \; {\underline{\boxed{\orange{\pmb{\frak{ Semi - Perimeter = 12 \; cm }}}}}}}}$

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~ Calculating the Area :

${\dashrightarrow{\qquad{\sf{ Area = \sqrt{s (s - a)(s - b)(s - c)} }}}} \\ \\ \\ \ {\dashrightarrow{\qquad{\sf{ Area = \sqrt{12 (12 - 10)(12 - 8)(12 - 6)} }}}} \\ \\ \\ \ {\dashrightarrow{\qquad{\sf{ Area = \sqrt{12 \times 2 \times 4 \times 6 } }}}} \\ \\ \\ \ {\dashrightarrow{\qquad{\sf{ Area = \sqrt{2 \times 3 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 } }}}} \\ \\ \\ \ {\dashrightarrow{\qquad{\sf{ Area =2 \times 3 \times 2 \times 2 }}}} \\ \\ \\ \ {\dashrightarrow{\qquad{\sf{ Area = 6 \times 4 }}}} \\ \\ \\ \ {\qquad \; \; {\therefore \; {\underline{\boxed{\purple{\pmb{\frak{ Area = 24 \; {cm}^{2} }}}}}}}}$

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~ Calculating the Cost :

${\longmapsto{\qquad{\sf{ Cost = Area \times Rate }}}} \\ \\ \\ \ {\longmapsto{\qquad{\sf{ Cost = 24 \times 9 }}}} \; \; \; \; \; \bigg\lgroup {\pink{\sf{Rs. \; 1 = 100 \; Paise }}} \bigg\rgroup \\ \\ \\ \ {\longmapsto{\qquad{\sf{ Cost = 24 \times \dfrac{9}{100} }}}} \\ \\ \\ \ {\longmapsto{\qquad{\sf{ Cost = 24 \times \cancel\dfrac{9}{100} }}}} \\ \\ \\ \ {\longmapsto{\qquad{\sf{ Cost = 24 \times 0.09 }}}} \\ \\ \\ \ {\qquad \; \; {\therefore \; {\underline{\boxed{\red{\pmb{\frak{ Cost = ₹ \; 2.16 }}}}}}}}$

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❛❛ Cost of Painting the Triangular board is ₹ 2.16 . ❜❜

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Answer 2

Rough Diagram:-

$\setlength{\unitlength}{1 cm}\begin{picture}(0,0)\thicklines\qbezier(1, 0)(1,0)(3,3)\qbezier(5,0)(5,0)(3,3)\qbezier(5,0)(1,0)(1,0)\put(2.85,3.2){ \bf 6 \ cm(x) }\put(0.5,-0.3){ \bf 10 \ cm(z) }\put(5.2,-0.3){ \bf 8 \ cm(y) }\end{picture}$

Given :

• 6 cm (x)
• 8 cm (y)
• 10 cm (z)

To Find :

The cost of painting it at the rate of 9 paise per cm square.

Solution :

$✴ \rm{Semi - Perimeter} = \dfrac{x + y+ z}{2}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \ \dashrightarrow \dfrac{6 + 8 + 10}{2}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \ \dashrightarrow \cancel\dfrac{24}{2}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \dashrightarrow \rm12 \: cm$

$✴ \rm{Area \: of \: the \: triangle = \sqrt{s(s - a)(s - b)(s - c) } }$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \dashrightarrow \rm{ \sqrt{12(12 - 6)(12 - 8)(12 - 10)} }$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \dashrightarrow \rm{ \sqrt{12 \times (6) \times (4) \times (2)} }$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \dashrightarrow \rm{ \sqrt{12 \times 6 \times 4 \times 2} }$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \dashrightarrow \rm{ \sqrt{12 \times (3 \times 2)\times 4 \times 2} }$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \dashrightarrow \rm{ \sqrt{12 \times 3 \times 4 \times 2 \times 2} }$

$\: \: \: \: \: \: \: \: \: \: \: \dashrightarrow \rm{ \sqrt{12 \times 12 \times 2 \times 2} }$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \dashrightarrow \rm{ \sqrt{(12)^{2} \times ( 2)^{2} } }$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \dashrightarrow \rm{ 12\times 2 = 24 \: cm ^{2} }$

$\begin{cases} \rm Cost \: of \: painting \: for \: area \: 1 \: cm ^{2} = ₹ 1 \: cm ^{2} \\ \\ \\ \rm{Cost \: of \: painting \: for \: area \: 24 \: {cm}^{2} = 0.09 \times 24 = ₹2.16 }\end{cases}$

∴Cost of painting for area 24 cm² = ₹2.16