Question

If the ratio of length, breadth and height of cuboid is 1:2:3. And the volume is 2058 cu. cm. Find :-
Length and breadth and height
TSA
LSA

Answer 1

$\huge{\bf{\green{\mathfrak{Question:-}}}}$

• If the ratio of length, breadth and height of cuboid is 1:2:3. And the volume is 2058 cm³.
• Find :-
• Length, breadth and height of the cuboid
• TSA
• LSA

$\huge {\bf{\orange{\mathfrak{Answer:-}}}}$

• $\textsf{Ratio of length, breadth and height is 1:2:3.}$

• $\sf{Volume\: of\: the \:cuboid =\: 2058 \:cm^{3}.}$

• $\textsf{Let the length, breadth and height of the cuboid be x.}$

• $\therefore{\sf{Length \: = \: x}}$

• $\therefore{\sf{Breadth \: = \: 2 \: * \: x \: = \: 2x}}$

• $\therefore{\sf{Height \: = \: 3 \: * \: x \: = \: 3x}}$

• $\boxed{\sf{Volume \: of \: the \: cuboid \: = \: lbh \: units^{3}.}}$

• $\boxed{\sf{x \: *\: 2x\: *\: 3x \: = \: 2058}}$

• $\sf{6x^{3} \: = \: 2058}$

• $\sf{x^{3} \: = \: \dfrac{2058}{6}}$

• $\sf{x^{3} \: = \: 343}$

• $\sf{x \: = \: \sqrt[3]{343}}$

• $\boxed{\sf{x \: = \: 7.}}$

• $\boxed{\textsf{Length of the cuboid = x = 7 cm.}}$

• $\boxed{\textsf{Breadth of the cuboid = 2x = 2 * 7 = 14 cm.}}$

• $\boxed{\textsf{Height of the cuboid = 3x = 3 * 7 = 21 cm.}}$

• $\boxed{\sf{TSA \:of\: the\: cuboid \:=\: 2(lb \:+ \:bh \:+ \:hl) \:units^{2}.}}$

• $\sf{2(7 \: * \: 14 \: + \: 14 \: * \: 21 \: + \: 21 \: * \: 7)}$

• $\sf{2(98 \: + \: 294 \: + \: 147)}$

• $\sf{2(539)}$

• $\boxed{\sf{1078 \: cm^{2}.}}$

• $\boxed{\sf{LSA\: of \:the \:cuboid \:=\: 2h(l \: + \: b) \: units^{2}.}}$

• $\sf{2 \: * \: 21(7 \: + \: 14)}$

• $\sf{42(21)}$

• $\boxed{\sf{882 \: cm^{2}.}}$

$\huge{\bf{\red{\mathfrak{Conclusion:-}}}}$

• $\boxed{\textsf{Length of the cuboid = 7 cm.}}$
• $\boxed{\textsf{Breadth of the cuboid = 2 * 7 = 14 cm.}}$

• $\boxed{\textsf{Height of the cuboid = 3 * 7 = 21 cm.}}$

• $\boxed{\sf{TSA \: of \: the \: cuboid \: = \: 1078 \: cm^{2}.}}$

• $\boxed{\sf{LSA \: of \: the \: cuboid \: = \: 882 \: cm^{2}.}}$

$\huge{\bf{\purple{\mathfrak{Formulas \: used:-}}}}$

• $\sf{TSA \:of\: the\: cuboid \:=\: 2(lb \:+ \:bh \:+ \:hl) \:units^{2}.}$

• $\sf{LSA\: of \:the \:cuboid \:=\: 2h(l \: + \: b) \: units^{2}.}$

• $\sf{Volume \: of \: the \: cuboid \: = \: lbh \: units^{3}.}$

Answer 2

Answer:

Given :-

• The ratio of length, breadth and height of cuboid is 1 : 2 : 3 and the volume is 2058 cm³.

To Find :-

• What is the length, breadth, height, TSA (Total surface area) and LSA (Lateral surface area).

Formula Used :-

$\sf\boxed{\bold{\pink{Volume\: of\: cuboid =\: Length \times Breadth \times Height}}}$

$\sf\boxed{\bold{\pink{T.S.A\: of\: Cuboid =\: 2(LB + BH + HL)}}}$

where,

• L = Length
• B = Breadth
• H = Height

$\sf\boxed{\bold{\pink{L.S.A\: of\: Cuboid =\: 2H(L + B)}}}$

Solution :-

Let, the length be x cm.

Breadth be 2x.

And, the height will be 3x.

According to the question by using the formula we get,

↦ $\sf x \times 2x \times 3x =\: 2058$

↦ $\sf 6{x}^{3} =\: 2058$

↦ $\sf {x}^{3} =\: \dfrac{\cancel{2058}}{\cancel{6}}$

↦ $\sf {x}^{3} =\: 343$

↦ $\sf x =\: \sqrt[3]{343}$

➦ $\sf\bold{x =\: 7\: cm}$

Hence, the required length, breadth and height of cuboid are :

✧ Length of cuboid :

↦ $\sf x\: cm$

➠ $\sf\bold{\green{7\: cm}}$

✧ Breadth of cuboid :

↦ $\sf 2x\: cm$

↦ $\sf 2 \times 7\: cm$

➠ $\sf\bold{\green{14\: cm}}$

And,

✧ Height of cuboid :

↦ $\sf 3x\: cm$

↦ $\sf 3 \times 7\: cm$

➠ $\sf\bold{\green{21\: cm}}$

$\therefore$ The length, breadth and height of cuboid is 7 cm, 14 cm and 21 cm respectively.

Now, we have to find the TSA or cuboid :

Given :

• Length = 7 cm
• Breadth = 14 cm
• Height = 21 cm

According to the question by using the formula we get,

$\implies \sf T.S.A\: of\: Cuboid =\: 2(7 \times 14 + 14 \times 21 + 21 \times 7)$

$\implies \sf T.S.A\: of\: Cuboid =\: 2(98 + 294 + 147)$

$\implies \sf T.S.A\: of\: Cuboid =\: 2(539)$

$\implies \sf T.S.A\: of\: Cuboid =\: 2 \times 538$

$\implies \sf\bold{\green{T.S.A\: of\: Cuboid =\: 1078\: {cm}^{2}}}$

$\therefore$ The total surface area or TSA of cuboid is 1078 cm².

Again, we have to find the LSA or lateral surface area of cuboid :

Given :

• Length = 7 cm
• Breadth = 14 cm
• Height = 21 cm

According to the question by using the formula we get,

$\implies \sf L.S.A\: of\: Cuboid =\: 2 \times 21(7 + 14)$

$\implies \sf L.S.A\: of\: Cuboid =\: 42(21)$

$\implies \sf L.S.A\: of\: Cuboid =\: 42 \times 21$

$\implies \sf\bold{\green{L.S.A\: of\: Cuboid =\: 882\: {cm}^{2}}}$

$\therefore$ The LSA or lateral surface area of cuboid is 882 cm².