a cubical block of side 7 cm is surmounted by a hemisphere. what is the greatest diameter the hemisphere can have? find the surface area of the solid.
What is the greatest diameter the hemisphere can have and what is the surface area of the solid?
Answer:
To determine the greatest diameter of the hemisphere placed on top of a cubical block with a side length of 7 cm, we need to find the side length of the cube that fits exactly inside the hemisphere. This side length would also represent the diameter of the hemisphere.
Let’s calculate this step by step:
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The diameter of the hemisphere is equal to the side length of the cube, as the cube fits perfectly inside the hemisphere. So, the diameter of the hemisphere is 7 cm.
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The total surface area of the solid (cube + hemisphere) can be calculated by adding the surface area of the cube and the surface area of the hemisphere.
Surface Area of Cube:
The formula for the surface area of a cube is 6a^2, where (a) is the side length of the cube.
Given that the side length of the cube (a = 7) cm:
Surface Area of Cube = (6 \times 7^2 = 294) cm².
Surface Area of Hemisphere:
The formula for the surface area of a hemisphere is 2\pi r^2 + \pi r^2, where (r) is the radius of the hemisphere (half of its diameter).
Given that the radius of the hemisphere is half of its diameter (7 cm), so the radius (r = \frac{7}{2} = 3.5) cm.
Surface Area of Hemisphere = (2\pi \times 3.5^2 + \pi \times 3.5^2 = 38.5\pi) cm².
Total Surface Area:
Adding both surface areas together:
Total Surface Area = Surface Area of Cube + Surface Area of Hemisphere = (294 + 38.5\pi \approx 408.93) cm².
Therefore, the greatest diameter the hemisphere can have on top of the cubical block is 7 cm, and the total surface area of the solid (cube + hemisphere) is approximately 408.93 cm².