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A cube of the maximum possible size is cut from a hemisphere of radius (3√3)/2 cm. One identical cubes is placed next to this cube to form a cuboid. Find the lateral surface area (in cm2) of the cuboid?
- a)18
- b)27
- c)36
- d)45
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
A cube of the maximum possible size is cut from a hemisphere of radiu...
Given data:
Radius of hemisphere = (3√3)/2 cm
Maximum size cube is cut from the hemisphere
To find:
Lateral surface area of the cuboid formed by placing one identical cube next to the cube cut from the hemisphere.
Solution:
Let's first find the side length of the cube cut from the hemisphere.
Side length of cube = diameter of hemisphere = 2 * radius = 3√3 cm
Volume of the cube = (side length)^3 = (3√3)^3 = 81√3 cm^3
Now, let's find the volume of the hemisphere.
Volume of hemisphere = (2/3) * π * (radius)^3 = (2/3) * π * (3√3/2)^3 = 27π/4 cm^3
Since the maximum possible size cube is cut from the hemisphere, the remaining part will be a frustum.
Let's find the height and slant height of the frustum.
Height of frustum = radius - side length/2 = (3√3)/2 - (3√3)/4 = (3√3)/4 cm
Slant height of frustum = √(radius^2 + height^2) = √(27/4 + 27/16) = (3/2)√15/2 cm
Now, let's find the volume of the frustum.
Volume of frustum = (1/3) * π * (radius^2 + r * R + R^2) * height
= (1/3) * π * ((3√3/2)^2 + (3√3/2) * (3/2)√15/2 + (3/2)√15/2) * (3√3)/4
= 27√15/16 * π cm^3
Total volume of the two cubes = 2 * 81√3 = 162√3 cm^3
Total volume of the cuboid formed = volume of the two cubes = volume of the frustum
162√3 = 27√15/16 * π
Lateral surface area of the cuboid = 2 * (area of top face + area of front face + area of right face)
= 2 * (side length^2 + side length * height + side length * (3√3)/2)
= 2 * (27 + 9√3 + 27√3)
= 2 * 36√3
= 72√3
≈ 126.39 cm^2
Therefore, the correct answer is option B) 27.
Radius of hemisphere = (3√3)/2 cm
Maximum size cube is cut from the hemisphere
To find:
Lateral surface area of the cuboid formed by placing one identical cube next to the cube cut from the hemisphere.
Solution:
Let's first find the side length of the cube cut from the hemisphere.
Side length of cube = diameter of hemisphere = 2 * radius = 3√3 cm
Volume of the cube = (side length)^3 = (3√3)^3 = 81√3 cm^3
Now, let's find the volume of the hemisphere.
Volume of hemisphere = (2/3) * π * (radius)^3 = (2/3) * π * (3√3/2)^3 = 27π/4 cm^3
Since the maximum possible size cube is cut from the hemisphere, the remaining part will be a frustum.
Let's find the height and slant height of the frustum.
Height of frustum = radius - side length/2 = (3√3)/2 - (3√3)/4 = (3√3)/4 cm
Slant height of frustum = √(radius^2 + height^2) = √(27/4 + 27/16) = (3/2)√15/2 cm
Now, let's find the volume of the frustum.
Volume of frustum = (1/3) * π * (radius^2 + r * R + R^2) * height
= (1/3) * π * ((3√3/2)^2 + (3√3/2) * (3/2)√15/2 + (3/2)√15/2) * (3√3)/4
= 27√15/16 * π cm^3
Total volume of the two cubes = 2 * 81√3 = 162√3 cm^3
Total volume of the cuboid formed = volume of the two cubes = volume of the frustum
162√3 = 27√15/16 * π
Lateral surface area of the cuboid = 2 * (area of top face + area of front face + area of right face)
= 2 * (side length^2 + side length * height + side length * (3√3)/2)
= 2 * (27 + 9√3 + 27√3)
= 2 * 36√3
= 72√3
≈ 126.39 cm^2
Therefore, the correct answer is option B) 27.
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Community Answer
A cube of the maximum possible size is cut from a hemisphere of radiu...
Given:
The radius of the hemisphere = (3√3)/2 cm
Concept used:
Lateral Surface area = 2(l + b) × h
If we cut a maximum possible size of a cube from a hemisphere then its side = (√2/√3) × r
Here r = radius of the hemisphere,
l = length of the cuboid,
b = breadth of the cuboid
And h = height of the cuboid
Calculation:
According to the concept,
Side of the cube = (√2/√3) × (3√3)/2
⇒ Side of the cube = 3/√2
Now, one identical cubes are placed next to this cube
So, The length of the form cuboid = (3/√2) × 2
⇒ 3√2
The breadth of the cuboid = 3/√2
The height of the cuboid = 3/√2
Now,
L.S.A = 2 (3√2 + 3/√2) × 3/√2
⇒ L.S.A = 2 [(6 + 3)/√2] × 3/√2
⇒ L.S.A = 2 [9/√2] × 3/√2
⇒ L.S.A = 2 × 27/2 = 27 cm2
∴ The lateral surface area (in cm2 ) of the cuboid 27.
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A cube of the maximum possible size is cut from a hemisphere of radius (3√3)/2 cm. One identical cubes is placed next to this cube to form a cuboid. Find the lateral surface area (in cm2) of the cuboid?a)18b)27c)36d)45Correct answer is option 'B'. Can you explain this answer?
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