
2 cubes of volume \[64 {\rm{c}}{{\rm{m}}^3}\] are joined end to end. Find the surface area of the resulting cuboid.
Answer
484.2k+ views
Hint: Here, we need to find the surface area of the resulting cuboid. First, we will find the length of the side of the two cubes. Then, we will find the dimensions of the resulting cuboid when the two cubes are joined end to end. We will then use the formula for total surface area of a cuboid and simplify it to find the required answer.
Formula used: We will use the following formula to solve the questions:
The volume of a cube is given by the formula \[V = {a^3}\], where \[a\] is the length of the side of the cube.
The total surface area of a cuboid is given by the formula \[2\left( {lb + bh + lh} \right)\], where \[l\] is the length, \[b\] is the breadth, and \[h\] is the height.
Complete step-by-step answer:
First, we will find the length of the side of the cubes using the formula for volume of a cube.
The volume of a cube is given by the formula \[V = {a^3}\], where \[a\] is the length of the side of the cube.
Substituting \[V = 64{\rm{ c}}{{\rm{m}}^3}\] in the formula, we get
\[ \Rightarrow 64 = {a^3}\]
Taking cube root of both sides, we get
\[\begin{array}{l} \Rightarrow \sqrt[3]{{64}} = \sqrt[3]{{{a^3}}}\\ \Rightarrow a = 4{\rm{ cm}}\end{array}\]
Therefore, the length of the side of the cube is 4 cm.
Now, let us draw the diagram to show how the resulting cuboid looks.
We can observe that when the two cubes are joined together end to end, the breadth and height of the resulting cuboid is 4 cm.
The length of the cuboid is the sum of the lengths of the sides of the two cubes.
Therefore, the length of the cuboid \[ = 4 + 4 = 8{\rm{ cm}}\].
Now, we will find the surface area of the resulting cuboid.
The total surface area of a cuboid is given by the formula \[2\left( {lb + bh + lh} \right)\], where \[l\] is the length, \[b\] is the breadth, and \[h\] is the height.
Substituting \[l = 8\] cm, \[b = 4\] cm, and \[h = 4\] cm in the formula, we get
Surface area of the resulting cuboid \[ = 2\left( {8 \times 4 + 4 \times 4 + 8 \times 4} \right){\rm{ c}}{{\rm{m}}^3}\]
Multiplying the terms in the parentheses, we get
Surface area of the resulting cuboid \[ = 2\left( {32 + 16 + 32} \right){\rm{ c}}{{\rm{m}}^3}\]
Adding the terms in the parentheses, we get
Surface area of the resulting cuboid \[ = 2\left( {80} \right){\rm{ c}}{{\rm{m}}^3}\]
Multiplying 2 by 80, we get
Surface area of the resulting cuboid \[ = 160{\rm{ c}}{{\rm{m}}^3}\]
Therefore, the surface area of the resulting cuboid is 160 cubic centimetres.
Note: We need to keep in mind that the words “surface area” refer to the total surface area, and not the curved/lateral surface area, unless specified. We might make a mistake by calculating the curved surface area instead of the total surface area using the formula \[2h\left( {l + b} \right)\].
We need to also understand that the \[64 {\rm{c}}{{\rm{m}}^3}\] is the volume of both the cube and not the volume of just one cube. If we take it as the volume of one cume then it will be wrong.
Formula used: We will use the following formula to solve the questions:
The volume of a cube is given by the formula \[V = {a^3}\], where \[a\] is the length of the side of the cube.
The total surface area of a cuboid is given by the formula \[2\left( {lb + bh + lh} \right)\], where \[l\] is the length, \[b\] is the breadth, and \[h\] is the height.
Complete step-by-step answer:
First, we will find the length of the side of the cubes using the formula for volume of a cube.
The volume of a cube is given by the formula \[V = {a^3}\], where \[a\] is the length of the side of the cube.
Substituting \[V = 64{\rm{ c}}{{\rm{m}}^3}\] in the formula, we get
\[ \Rightarrow 64 = {a^3}\]
Taking cube root of both sides, we get
\[\begin{array}{l} \Rightarrow \sqrt[3]{{64}} = \sqrt[3]{{{a^3}}}\\ \Rightarrow a = 4{\rm{ cm}}\end{array}\]
Therefore, the length of the side of the cube is 4 cm.
Now, let us draw the diagram to show how the resulting cuboid looks.

We can observe that when the two cubes are joined together end to end, the breadth and height of the resulting cuboid is 4 cm.
The length of the cuboid is the sum of the lengths of the sides of the two cubes.
Therefore, the length of the cuboid \[ = 4 + 4 = 8{\rm{ cm}}\].
Now, we will find the surface area of the resulting cuboid.
The total surface area of a cuboid is given by the formula \[2\left( {lb + bh + lh} \right)\], where \[l\] is the length, \[b\] is the breadth, and \[h\] is the height.
Substituting \[l = 8\] cm, \[b = 4\] cm, and \[h = 4\] cm in the formula, we get
Surface area of the resulting cuboid \[ = 2\left( {8 \times 4 + 4 \times 4 + 8 \times 4} \right){\rm{ c}}{{\rm{m}}^3}\]
Multiplying the terms in the parentheses, we get
Surface area of the resulting cuboid \[ = 2\left( {32 + 16 + 32} \right){\rm{ c}}{{\rm{m}}^3}\]
Adding the terms in the parentheses, we get
Surface area of the resulting cuboid \[ = 2\left( {80} \right){\rm{ c}}{{\rm{m}}^3}\]
Multiplying 2 by 80, we get
Surface area of the resulting cuboid \[ = 160{\rm{ c}}{{\rm{m}}^3}\]
Therefore, the surface area of the resulting cuboid is 160 cubic centimetres.
Note: We need to keep in mind that the words “surface area” refer to the total surface area, and not the curved/lateral surface area, unless specified. We might make a mistake by calculating the curved surface area instead of the total surface area using the formula \[2h\left( {l + b} \right)\].
We need to also understand that the \[64 {\rm{c}}{{\rm{m}}^3}\] is the volume of both the cube and not the volume of just one cube. If we take it as the volume of one cume then it will be wrong.
Recently Updated Pages
The correct geometry and hybridization for XeF4 are class 11 chemistry CBSE

Water softening by Clarks process uses ACalcium bicarbonate class 11 chemistry CBSE

With reference to graphite and diamond which of the class 11 chemistry CBSE

A certain household has consumed 250 units of energy class 11 physics CBSE

The lightest metal known is A beryllium B lithium C class 11 chemistry CBSE

What is the formula mass of the iodine molecule class 11 chemistry CBSE

Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Given that HCF 306 657 9 find the LCM 306 657 class 9 maths CBSE

The highest mountain peak in India is A Kanchenjunga class 9 social science CBSE

A piece of wire 20 cm long is bent into the form of class 9 maths CBSE

Difference Between Plant Cell and Animal Cell

What is the difference between Atleast and Atmost in class 9 maths CBSE
