Question

The lateral surface area of a solid rectangular box is 520 cm2 and its length , breadth and height are in the ratio 9:4:5. Find the total surface area of the box.​

Answer 1

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Answer 2

Given Question :-

• The lateral surface area of a solid rectangular box is 520 cm² and its length , breadth and height are in the ratio 9:4:5. Find the total surface area of the box.

ANSWER

Given :-

• The lateral surface area of a solid rectangular box is 520 cm².

• Length : Breadth : Height = 9 : 4 : 5.

To Find :-

• The total surface area of the box.

$\begin{gathered}\Large{\bold{{\underline{Formula\:Used - }}}} \end{gathered}$

$\boxed{ \bf \: Curved Surface Area_{(cuboid)} = 2 \times (l + b) \times h}$

$\boxed{ \bf \: Total \: Surface \: Area_{(cuboid)} = 2(lb + bh + hl}$

where,

• l = Length of Cuboid

• b = Breadth of Cuboid

• h = Height of Cuboid

$\large\underline{\bold{Solution :- }}$

Given that

• Length : Breadth: Height = 9 : 4 : 5

$\begin{gathered}\begin{gathered}\bf \:Let -\begin{cases} &\sf{Length = 9x} \\ &\sf{Breadth = 4x}\\ &\sf{Height = 5x} \end{cases}\end{gathered}\end{gathered}$

Also,

• Curved Surface Area of box = 520 cm²

We know,

• Curved Surface Area of box is given by

$\rm :\longmapsto\:Curved \: Surface \: Area_{(box)} = 2 \times (l + b) \times h$

• On substituting the values, we get

$\rm :\longmapsto\:520 = 2(9x + 4x) \times 5x$

$\rm :\longmapsto\:520 = 10x \times 13x$

$\rm :\longmapsto\:130 {x}^{2} = 520$

$\rm :\implies\: {x}^{2} = 4$

$\bf\implies \:x \: = \: 2$

Hence,

• Dimensions of box are as

$\begin{gathered}\begin{gathered}\bf \:Dimensions -\begin{cases} &\sf{Length = 9x = 18 \: cm} \\ &\sf{Breadth = 4x = 8 \: cm}\\ &\sf{Height = 5x = 10 \: cm} \end{cases}\end{gathered}\end{gathered}$

Now,

We know,

• Total Surface Area of Box is given by

$\rm :\longmapsto\:Total \: Surface \: Area_{(box)} = 2(lb + bh + hl)$

$\rm :\longmapsto\:2(18 \times 8 + 8 \times 10 + 10 \times 18)$

$\rm :\longmapsto\:2(144 + 80 + 180)$

$\rm :\longmapsto\:2 \times 404$

$\rm :\longmapsto\:808 \: {cm}^{2}$

$\rm :\implies\: \boxed{ \bf \: Total \: Surface \: Area_{(box)} = 808 \: {cm}^{2} }$

Additional Information

Cube: 

• A cube is a three-dimensional shape which is defined XYZ plane. It has six faces, eight vertices and twelve edges. All the faces of the cube are in square shape and have equal dimensions.

Cuboid: 

• A cuboid is also a polyhedron having six faces, eight vertices and twelve edges. The faces of the cuboid are parallel. But not all the faces of a cuboid are equal in dimensions.

Formula's of Cube :-

• Total Surface Area = 6(side)²

• Curved Surface Area = 4(side)²

• Volume of Cube = (side)³

• Diagonal of a cube = √3(side)

• Perimeter of cube = 12 x side

Formula's of Cuboid :-

• Total Surface area = 2 (Length x Breadth + breadth x height + Length x height)

• Curved Surface area = 2 height(length + breadth)

• Volume of the cuboid = (length × breadth × height)

• Diagonal of the cuboid =√(l² + b² + h²)

• Perimeter of cuboid = 4 (length + breadth + height)