Question

the whole surface of a cuboid is 214cm square , volume is 210cm cube and the area of the base is 42cm square . find its , edge.​

Answer 1

Answer:

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Step-by-step explanation:

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Answer 2

Given:

• Total surface area of a cuboid = 214 cm²
• Volume of cuboid = 210 cm³
• Area of base of cuboid = 42 cm²

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To find:

• Dimensions of cuboid.

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Solution:

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$\underline{\bigstar\:\boldsymbol{As\:per\:given\: Question\::}}$

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$\star\;\sf Area\;of\;base = l \times b$

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$:\implies\sf l \times b = 42$

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$:\implies\bf lb = 42\;cm^2\;\;\;\;\;\;\;\;\;\bigg\lgroup\bf eq.\;(1) \bigg\rgroup$

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${\underline{\frak{\dag\;We\;know\;that\;:}}}$

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$\star\;{\boxed{\bf{\purple{Volume_{\;(cuboid)} = l \times b \times height}}}}$

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${\underline{\frak{\dag\;Putting\;values\;:}}}$

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$:\implies\sf 210 = l \times b \times h$

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$:\implies\sf 210 = 42 \times h\;\;\;\;\;\;\;\;\;\bigg\lgroup\bf \because\; l \times b = 42 \bigg\rgroup$

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$:\implies\sf h = \dfrac{210}{42}$

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$:\implies\sf h = \cancel{ \dfrac{210}{42}}$

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$:\implies{\underline{\boxed{\bf{\purple{5\;cm}}}}}\;\bigstar$

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Given that,

Total surface area of a cuboid = 214 cm²

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${\underline{\frak{\dag\;We\;know\;that\;:}}}$

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$\star\;{\boxed{\bf{\pink{TSA_{\;(cuboid)} = 2(lb + bh + hl)}}}}$

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${\underline{\frak{\dag\;Putting\;values\;:}}}$

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$:\implies\sf 2(l \times b + b \times 5 + 5 \times l) = 214$

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$:\implies\sf 2(lb + 5b + 5l) = 214$

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$:\implies\sf 2(lb + 5b + 5l) = 214$

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$:\implies\sf 2(l + b)5 + 2lb = 214$

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$:\implies\sf 10(l + b) + 2 \times 42 = 214\;\;\;\;\;\;\;\;\;\bigg\lgroup\bf \because\;lb = 42 \bigg\rgroup$

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$:\implies\sf 10(l + b) + 84 = 214$

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$:\implies\sf 10(l + b) = 214 - 84$

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$:\implies\sf 10(l + b) = 214 - 84$

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$:\implies\sf 10(l + b) = 130$

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$:\implies\sf l + b = \dfrac{130}{10}$

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$:\implies\sf l + b = \cancel{ \dfrac{130}{10}}$

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$:\implies\sf l + b = 13$

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$:\implies\bf l = (13 - b)\;\;\;\;\;\;\;\;\;\bigg\lgroup\bf eq.\;(2) \bigg\rgroup$

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${\underline{\sf{\bigstar\;Now,\;Putting\;eq.(2)\;in\;eq.(1)\;:}}}$

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$:\implies\sf (13 - b)b = 42$

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$:\implies\sf 13b - b^2 = 42$

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$:\implies\sf b^2 - 13b + 42 = 0$

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$\;\;\;\;{\underline{\frak{\dag\; Splitting\;middle\;term\;:}}}$

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$:\implies\sf b^2 - 7b - 6b + 42 = 0$

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$:\implies\sf b(b - 7) -6(b - 7) = 0$

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$:\implies\sf (b - 7)(b - 6) = 0$

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$:\implies\bf b = 6,7$

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${\underline{\sf{\bigstar\;Now,\; Substituting\;values\;of\;b\;in\;eq.(2)\;:}}}$

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$\;\;{\underline{\sf{\dag\;At\;b = 6\;:}}}$

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$:\implies\sf l = 13 - 6$

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$:\implies\sf l = 7$

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$\;\;{\underline{\sf{\dag\;At\;b = 7\;:}}}$

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$:\implies\sf l = 13 - 7$

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$:\implies\sf l = 6$

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We know that, Length is always greater than breadth.

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Therefore,

• Length of Cuboid = 7 cm
• Breadth of cuboid = 6 cm
• Height of cuboid = 5 cm

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$\therefore$ Dimensions of Cuboid is 7 cm × 6 cm × 5 cm.