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: Understand the Problem:
We are given that each edge of the Rubik's cube measures 6 cm, and we need to find the length of the diagonal of the cube. The formula to calculate the diagonal of a cube is: Diagonal = a√3
where a is the length of the edge of the cube.
Step 2: Substituting the Value of the Edge Length:
Given that the edge length (a) is 6 cm, substitute this value into the formula: Diagonal = 6√3
Final Answer:
The length of the diagonal of the Rubik's cube is 6√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: Understand the Problem:
We are given the length of the edge of a Rubik's cube as 7 cm, and we need to find its volume. The formula for the volume of a cube is: Volume = a³
where a is the length of the edge of the cube.
Step 2: Substituting the Value of the Edge Length:
Given that the edge length (a) is 7 cm, substitute this value into the formula: Volume = 7³ Volume = 7 × 7 × 7 Volume = 343 cm³
Final Answer:
The volume of the Rubik's cube is 343 cm³.
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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: Understand the Problem:
We are given a hemisphere with a base radius of 7 cm, and we are required to find its curved surface area. The formula for the curved surface area of a hemisphere is given by: Curved Surface Area = 2πr²
Step 2: Substituting the Value of the Radius:
Given that the radius (r) is 7 cm, substitute this value into the formula: Curved Surface Area = 2π(7)² Curved Surface Area = 2π(49) Curved Surface Area = 98π
Step 3: Approximating π:
Using the approximation π ≈ 3.14, we get: Curved Surface Area = 98 × 3.14 Curved Surface Area = 307.72 cm²
Final Answer:
The curved surface area of the hemisphere is approximately 308 cm².
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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: Understand the Problem:
We are given two cubes, each with an edge of 4 cm. When they are joined end-to-end, they form a cuboid. We need to find the surface area of the resulting cuboid.
Step 2: Dimensions of the Resulting Cuboid:
Since the cubes are joined end-to-end, the length of the resulting cuboid will be the sum of the edges of the two cubes.
Thus, the dimensions of the cuboid will be:
Length = 4 + 4 = 8 cm
Width = 4 cm
Height = 4 cm
Step 3: Formula for Surface Area of a Cuboid:
The surface area of a cuboid is given by the formula: Surface Area = 2 * (Length * Width + Width * Height + Height * Length)
Step 4: Substituting the Values:
Substitute the dimensions of the cuboid into the formula: Surface Area = 2 * (8 * 4 + 4 * 4 + 4 * 8) Surface Area = 2 * (32 + 16 + 32) Surface Area = 2 * 80 Surface Area = 160 cm²
Final Answer:
The surface area of the resulting cuboid is 160 cm².
