Question

the total surface area is a cuboid is 3328 m square if its dimensions are in the ratio 4 ratio 3 ratio 2 find its volume

Answer 1

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Answer 2

$\huge{ \underline{ \overline{ \mid{ \mathfrak{ \purple{ \: \: SOLUTION \: \: } \mid}}}}}$

$\underline{ \mathfrak{ \: Given, \: }} \\ \\ \mathtt{Surface \: \: area \: \: of \: \: the \: cuboid = 3328 \: m {}^{3} } \\ \\ \text{Dimentions of the cuboid are in } \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \text{the ratio = 4 : 3 : 2} \\ \\ \\ \underline{ \mathfrak{ \: \: Suppose, \: \: }} \\ \\ : \rightsquigarrow \: \text{The length of the cuboid = 4x} \\ \\ : \rightsquigarrow \: \text{The breadth of the cuboid = 3x} \\ \\ : \rightsquigarrow \: \text{The heigth of the cuboid = 2x} \\ \\ \\ \underline{ \mathfrak{ \: \: To \: \: find :- \: \: }} \\ \\ \: \: \: \: \: \: \: \text{ Volume \: of \: the \: cuboid = ?}$

$\underline{ \mathfrak{ \: \: We \: \: know \: \: that, \: \: }} \\ \\ \boxed{ \bold{ \: </p><p></p><p>Surface \: \: area \: \: of \: \: the \: \: cuboid = 2 (lb + bh + hl)}}$

$\bold{ \underline{ \: \: A.T.Q., \: \: }} \\ \\ \: \: \: \: \: \: \: \mathtt{2(4x.3x+3x.2x+2x.4x) = 3328} \\ \\ \mathtt{\Longrightarrow \: 2(12x {}^{2} + 6x {}^{2} + 8x {}^{2} ) = 3328} \\ \\ \mathtt{\Longrightarrow \: 2 \times 26x {}^{2} = 3328} \\ \\ \mathtt{\Longrightarrow \: 52x {}^{2} = 3328} \\ \\ \mathtt{\Longrightarrow \: x {}^{2} = \frac{3328}{52} } \\ \\ \mathtt{\Longrightarrow \: x {}^{2} = 64} \\ \\ \mathtt{\Longrightarrow \: x = \sqrt{64} } \\ \\ \: \: \: \: \mathtt{ \therefore \: \:x = \pm \: 8 } \\ \\ \mathtt{ \: \: \: \: \: As \: \: the \: \: value \: \: of \: \: \red{x} \: \: can't \: \: be \: \: negative,} \\ \mathtt{so, \: \: \underline{ \: \red{ x = 8 }\: }}$

$\mathtt{ \therefore \: \:Length ,l= 4x =(4 \times 8 )\:m = 32 \: m} \\ \\ \mathtt{ \rightsquigarrow \: Breadth,b = 3x = (3 \times 8) \: m = 24 \: m} \\ \\ \mathtt{ \rightsquigarrow \: Height,h = 2x = (2 \times 8) \: m = 16 \: m}$

$\mathfrak{ \: Now,} \\ \\ \underline{ \mathfrak{ \: \: We \: \: know \: \: that, \: \: }} \\ \\ \boxed{ \bold{Volume \: \: of \: \: a \: \: cuboid = l \times b \times h}}$

$\underline{ \mathtt{ \: \: Putting \: \: the \: \: value \: \: of \: \: \red{ l }, \red{b} \: \: and \: \: \red{h }\: \: in \: \: the \: \:}} \\ \underline{ \mathtt{ \: \: formula, we \: \: get \: \: }} \\ \\ \mathtt{Volume =(32 \times 24 \times 16) \: \: m {}^{3} } \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \mathtt{ = 12288 \: \: m {}^{3} }$

$\underline{ \bold{ \: \:Volume \: \: of \: \: the \: \: cuboid = 12288 \: m {}^{3} \: \: }}$