Answer

Verified

454.8k+ views

Hint: First of all, find the volume of the 3 cubes of edges 3 cm, 4 cm, and 5 cm. Then equal the total volume of three cubes to the volume of the new cube to get the edge of the new cube. Then find the surface area of the cube, that is, \[S=6{{\left( \text{edge} \right)}^{2}}\].

Complete step-by-step answer:

Here, we are given 3 cubes whose edges measure 3 cm, 4 cm, and 5 cm are used to form a single cube. We have to find the edge as well as the surface area of the new cube.

Before proceeding with the question, we must know that when one object of a particular shape is converted into other objects/objects of the same or different shape, then volume always remains constant. That is, the volume of the object before and after the conversion would be the same, while the other quantities such as surface area, length, breadth, radius, etc. could change.

Now, we are given 3 cubes of edges 3 cm, 4 cm, and 5 cm. We will first find the volume of these cubes.

We know that the volume of the cube \[={{a}^{3}}....\left( i \right)\] where ‘a’ is the edge of the cube.

So, we get,

The volume of the cube of edge 5 cm \[={{\left( 5 \right)}^{3}}c{{m}^{3}}=125c{{m}^{3}}....\left( ii \right)\]

The volume of the cube of edge 4 cm \[={{\left( 4 \right)}^{3}}c{{m}^{3}}=64c{{m}^{3}}....\left( iii \right)\]

The volume of the cube of edge 3 cm \[={{\left( 3 \right)}^{3}}c{{m}^{3}}=27c{{m}^{3}}....\left( iv \right)\]

Now, we are given that these 3 cubes are used to form a new cube. Let us consider the edge of the new cube as A. So, we get,

The volume of the new cube \[={{\left( A \right)}^{3}}c{{m}^{3}}....\left( v \right)\]

As we already know that when one or more object is converted into another, then volume remains constant, therefore we get,

(Volume of the cube of edge 5 cm) + (Volume of the cube of edge 4 cm) + (Volume of the cube of edge 3 cm) = (Volume of the cube of edge A cm)

By substituting the values of LHS and RHS from the equation (ii), (iii), (iv) and (v), we get,

\[125+64+27={{A}^{3}}\]

Or, \[{{A}^{3}}=216\]

By taking cube root on both the sides, we get,

\[A=\sqrt[3]{216}\]

\[A=6\text{ cm}\]

Hence, the edge of the new cube formed is 6 cm.

Now, we know that the surface area of the cube \[=6{{\left( edge \right)}^{2}}\] or \[6{{a}^{2}}\]

By substituting a = A = 6 cm, we get,

The surface area of the cube \[=6{{\left( 6 \right)}^{2}}\]

\[=216\text{ c}{{\text{m}}^{2}}\]

Note: Students must remember that whenever a 3-dimensional object is converted into other objects, then volume always remains constant. Also, students must take care of the units like the volume of the object is always in cubic meters while the area of the object is always in square meters and so on in any question.

Complete step-by-step answer:

Here, we are given 3 cubes whose edges measure 3 cm, 4 cm, and 5 cm are used to form a single cube. We have to find the edge as well as the surface area of the new cube.

Before proceeding with the question, we must know that when one object of a particular shape is converted into other objects/objects of the same or different shape, then volume always remains constant. That is, the volume of the object before and after the conversion would be the same, while the other quantities such as surface area, length, breadth, radius, etc. could change.

Now, we are given 3 cubes of edges 3 cm, 4 cm, and 5 cm. We will first find the volume of these cubes.

We know that the volume of the cube \[={{a}^{3}}....\left( i \right)\] where ‘a’ is the edge of the cube.

So, we get,

The volume of the cube of edge 5 cm \[={{\left( 5 \right)}^{3}}c{{m}^{3}}=125c{{m}^{3}}....\left( ii \right)\]

The volume of the cube of edge 4 cm \[={{\left( 4 \right)}^{3}}c{{m}^{3}}=64c{{m}^{3}}....\left( iii \right)\]

The volume of the cube of edge 3 cm \[={{\left( 3 \right)}^{3}}c{{m}^{3}}=27c{{m}^{3}}....\left( iv \right)\]

Now, we are given that these 3 cubes are used to form a new cube. Let us consider the edge of the new cube as A. So, we get,

The volume of the new cube \[={{\left( A \right)}^{3}}c{{m}^{3}}....\left( v \right)\]

As we already know that when one or more object is converted into another, then volume remains constant, therefore we get,

(Volume of the cube of edge 5 cm) + (Volume of the cube of edge 4 cm) + (Volume of the cube of edge 3 cm) = (Volume of the cube of edge A cm)

By substituting the values of LHS and RHS from the equation (ii), (iii), (iv) and (v), we get,

\[125+64+27={{A}^{3}}\]

Or, \[{{A}^{3}}=216\]

By taking cube root on both the sides, we get,

\[A=\sqrt[3]{216}\]

\[A=6\text{ cm}\]

Hence, the edge of the new cube formed is 6 cm.

Now, we know that the surface area of the cube \[=6{{\left( edge \right)}^{2}}\] or \[6{{a}^{2}}\]

By substituting a = A = 6 cm, we get,

The surface area of the cube \[=6{{\left( 6 \right)}^{2}}\]

\[=216\text{ c}{{\text{m}}^{2}}\]

Note: Students must remember that whenever a 3-dimensional object is converted into other objects, then volume always remains constant. Also, students must take care of the units like the volume of the object is always in cubic meters while the area of the object is always in square meters and so on in any question.

Recently Updated Pages

What number is 20 of 400 class 8 maths CBSE

Which one of the following numbers is completely divisible class 8 maths CBSE

What number is 78 of 50 A 32 B 35 C 36 D 39 E 41 class 8 maths CBSE

How many integers are there between 10 and 2 and how class 8 maths CBSE

The 3 is what percent of 12 class 8 maths CBSE

Find the circumference of the circle having radius class 8 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

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

One cusec is equal to how many liters class 8 maths CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE

Change the following sentences into negative and interrogative class 10 english CBSE