Question

The perimeter of a triangular field is 540m and its sides are in a ratio 25 : 17 : 12. Find the area of the triangle. Also, find the cost of cultivating the field at Rs. 24.60 per 100m².​

Answer 1

$\frak{Given}\begin{cases}\sf{Perimeter\;of\;the\;field}=\bf{540\;m}\\\sf{Ratio\;of\;the\;sides}=\bf{25:17:12}\end{cases}$

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Need to find: The area of the triangle and the cost of cultivating the field at Rs. 24.60 per 100m².

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⧠ Let the sides be 25x, 17x, 12x.

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$\bigstar\;\boldsymbol{\underline{According\;to\;the\;question,}}$

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$\sf{25x+17x+12x=540}\\\\\sf{=54x=540}\\\\\sf{=x=\dfrac{\cancel{540}}{\cancel{54}}}\\\\\sf{=x=10}$

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∴ The sides are,

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• (25 × 10)m = 250m
• (17 × 10)m = 170m
• (12 × 10)m = 120m

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$\sf\therefore\:{s=\dfrac{sum\;of\;sides}{2}}\\\\\sf{=\dfrac{\cancel{540}}{\cancel{\:2\:}}}\\\\\sf{=270\:m}$

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Now, area of the triangular field,

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$\bigstar\:{\pink{\underline{\boxed{\sf{\sqrt{s(s-a)(s-b)(s-c)}}}}}}\\\\\sf{=\sqrt{270(270-250)(270-170)(270-120)}}\\\\\sf{=\sqrt{270×20×100×150}}\\\\\sf{=\sqrt{3×3×3×10×2×10×2×5×10×3×5×10}}\\\\\sf{=\sqrt{2×3×3×5×10×10\;m^2}}\\\\{\purple{\underline{\boxed{\frak{=9000\;m^2}}}}}{\:\bigstar}$

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Cost of cultivating the field at Rs. 24.60 per 100m²,

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$\sf{=Rs.\;\cfrac{9000×24.60}{100}}\\\\\\\sf{=Rs.\;\cfrac{9000×2460}{100×100}}\\\\\\={\blue{\underline{\boxed{\sf{Rs.\;{\frak{2214}}}}}}}{\:\bigstar}$

Answer 2

$\large \mathfrak{ \purple{ \underline{Given:}}}$

$\sf{ \star{ \: {Perimeter\;of\;a \: triangular \: field}={540\;m}}}$

$\sf{ \star{ \: Ratio\;of\;the\;sides}={25:17:12}}$

$\large \mathfrak{ \purple { \underline{To \: Find : }}}$

${\star}$ The area of the triangle.

${\star}$ The cost of cultivating the field at Rs. 24.60 per 100m².

${ \sf{☯ \: Let \: the \: sides \: be \: 25x, 17x, 12x.}}$

$\large \mathfrak{ \purple{\underline{According \: to \: the \: question,}}}$

$\begin{gathered}\sf{ \to{25x+17x+12x=540}}\\\\\sf{ \to54x=540}\\\\\sf{ \to \: x=\dfrac{\cancel{540}}{\cancel{54}}}\\\\\sf{ \to \: { \red{x=10}}}\end{gathered}$

${ \sf{∴ The \: sides \: are, }} \\ { \sf{༐ \: \: (25 × 10)m = 250m }}\\{ \sf{ ༐ \: \: (17 × 10)m = 170m}} \\{ \sf{༐ \: \: (12 × 10)m = 120m}}$

$\begin{gathered}\sf\therefore\:{s=\dfrac{sum\;of\;sides}{2}}\\\\\sf{=\dfrac{\cancel{540}}{\cancel{\:2\:}}}\\\\\sf{ \red{=270\:m}}\end{gathered}$

$\large \mathfrak{ \purple{ \underline{ Let's \: find \: the \: area \: of \: the \: triangular \: field,}}}$

$\begin{gathered}{\green{\boxed{\sf{\sqrt{s(s-a)(s-b)(s-c)}}}}}\\\\\sf{=\sqrt{270(270-250)(270-170)(270-120)}}\\\\\sf{=\sqrt{270×20×100×150}}\\\\\sf{=\sqrt{3×3×3×10×2×10×2×5×10×3×5×10}}\\\\\sf{=\sqrt{2×3×3×5×10×10\;m^2}}\\\ \\ \red{\boxed{\frak{=9000\;m^2}}}\end{gathered}$

$\large \mathfrak{ \purple{ \underline{Let's \: find \: the \: cost \: of \: cultivating \: the \: field \: at \: Rs. 24.60 \: per \: 100m²,}}}$

$\begin{gathered}\sf{ \to \: Rs.\;\cfrac{9000×24.60}{100}}\\\\\\\sf{ \to \: Rs.\;\cfrac{9000×2460}{100×100}}\\\\\\ \to{\red{ \boxed{\sf{Rs.\;{\frak{2214}}}}}}\end{gathered}$

Hᴇɴᴄᴇ, ᴛʜᴇ ᴀʀᴇᴀ ᴏғ ᴛʜᴇ ᴛʀɪᴀɴɢᴜʟᴀʀ ғɪᴇʟᴅ ɪs 9000 ᴍ² ᴀɴᴅ ᴛʜᴇ ᴄᴏsᴛ ᴏғ ᴄᴜʟᴛɪᴠᴀᴛɪɴɢ ᴛʜᴇ ғɪᴇʟᴅ ᴀᴛ Rs. 24.60 ᴘᴇʀ 100ᴍ² ɪs Rs. 2214.