On a Sunday your parents took you to a fair. You could see lot of toys displayed and you wanted them to buy a Rubik's cube and a strawberry ice-cream for you.
Question: 1
Find the length of the diagonal of Rubik's cube if each edge measures 6 cm.
Show Hint
Do not confuse the face diagonal (\( a\sqrt{2} \)) with the space diagonal (\( a\sqrt{3} \)).
Step 1: Understanding the Concept:
The diagonal of a cube connects two opposite corners through the center of the cube. Step 2: Key Formula or Approach:
Diagonal of a cube with side \( a \) is given by:
\[ \text{Diagonal} = a\sqrt{3} \] Step 3: Detailed Explanation:
Given edge length \( a = 6 \) cm.
Substituting in the formula:
\[ \text{Diagonal} = 6\sqrt{3} \text{ cm} \] Step 4: Final Answer:
The length of the diagonal is \( 6\sqrt{3} \) cm.
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Question: 2
Find the volume of Rubik's cube if the length of the edge is 7 cm.
Show Hint
Always include the correct unit of measurement. Volume is measured in cubic units (\( \text{cm}^3 \)).
Step 1: Understanding the Concept:
Volume represents the space occupied by the cube. Step 2: Key Formula or Approach:
Volume of cube \( = a^3 \). Step 3: Detailed Explanation:
Given edge \( a = 7 \) cm.
\[ \text{Volume} = 7 \times 7 \times 7 = 343 \text{ cm}^3 \] Step 4: Final Answer:
The volume of the cube is 343 \( \text{cm}^3 \).
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Question: 3
What is the curved surface area of hemisphere (ice-cream) if the base radius is 7 cm ?
Show Hint
Total Surface Area would be \( 3\pi r^2 \), but the question specifically asks for Curved Surface Area.
Step 1: Understanding the Concept:
Curved Surface Area (CSA) refers to the area of the outer curved part of the hemisphere. Step 2: Key Formula or Approach:
CSA of hemisphere \( = 2\pi r^2 \). Step 3: Detailed Explanation:
Given radius \( r = 7 \) cm. Use \( \pi = \frac{22}{7} \).
\[ \text{CSA} = 2 \times \frac{22}{7} \times 7 \times 7 \]
\[ \text{CSA} = 2 \times 22 \times 7 \]
\[ \text{CSA} = 44 \times 7 = 308 \text{ cm}^2 \] Step 4: Final Answer:
The curved surface area is 308 \( \text{cm}^2 \).
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Question: 4
If two cubes of edges 4 cm are joined end-to-end, then find the surface area of the resulting cuboid.
Show Hint
The area decreases because two faces (one from each cube) are now hidden in the joint.
Step 1: Understanding the Concept:
When two cubes are joined end-to-end, only the length changes while breadth and height remain the same. Step 2: Key Formula or Approach:
Surface Area of cuboid \( = 2(lb + bh + hl) \). Step 3: Detailed Explanation:
Dimensions of the new cuboid:
Length (\( l \)) = \( 4 + 4 = 8 \) cm.
Breadth (\( b \)) = 4 cm.
Height (\( h \)) = 4 cm.
\[ \text{Surface Area} = 2(8 \times 4 + 4 \times 4 + 4 \times 8) \]
\[ \text{Surface Area} = 2(32 + 16 + 32) \]
\[ \text{Surface Area} = 2(80) = 160 \text{ cm}^2 \] Step 4: Final Answer:
The surface area of the resulting cuboid is 160 \( \text{cm}^2 \).
