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A solid cylinder of diameter 12 cm and height 15 cm is melted and recast into 12 toys in the shape of a right circular cone mounted on a hemisphere. Find the radius of the hemisphere, if height of the cone is 3 times the radius.
- a)3 cm
- b)6 cm
- c)5 cm
- d)9 cm
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
A solid cylinder of diameter 12 cm and height 15 cm is melted and reca...
To solve this problem, we need to find the radius of the hemisphere when a solid cylinder of diameter 12 cm and height 15 cm is melted and recast into 12 toys in the shape of a right circular cone mounted on a hemisphere.
Let's break down the problem into smaller steps:
Step 1: Find the volume of the cylinder
The volume of a cylinder is given by the formula V = πr^2h, where r is the radius and h is the height. In this case, the diameter is given as 12 cm, so the radius (r) is half of that, which is 6 cm. The height (h) is given as 15 cm. Substituting these values into the formula, we get:
V_cylinder = π(6 cm)^2(15 cm)
V_cylinder = 540π cm^3
Step 2: Find the volume of one toy
The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius and h is the height. In this case, the height of the cone is given as 3 times the radius, so the height (h) is 3r. Substituting these values into the formula, we get:
V_cone = (1/3)πr^2(3r)
V_cone = πr^3 cm^3
Step 3: Find the volume of 12 toys
Since the melted cylinder is recast into 12 toys, the total volume of the toys should be equal to the volume of the cylinder. Therefore, we can set up the equation:
12V_cone = V_cylinder
12(πr^3) = 540π
r^3 = 45
Taking the cube root of both sides, we get:
r = 3 cm
Therefore, the radius of the hemisphere is 3 cm, which is option A.
Let's break down the problem into smaller steps:
Step 1: Find the volume of the cylinder
The volume of a cylinder is given by the formula V = πr^2h, where r is the radius and h is the height. In this case, the diameter is given as 12 cm, so the radius (r) is half of that, which is 6 cm. The height (h) is given as 15 cm. Substituting these values into the formula, we get:
V_cylinder = π(6 cm)^2(15 cm)
V_cylinder = 540π cm^3
Step 2: Find the volume of one toy
The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius and h is the height. In this case, the height of the cone is given as 3 times the radius, so the height (h) is 3r. Substituting these values into the formula, we get:
V_cone = (1/3)πr^2(3r)
V_cone = πr^3 cm^3
Step 3: Find the volume of 12 toys
Since the melted cylinder is recast into 12 toys, the total volume of the toys should be equal to the volume of the cylinder. Therefore, we can set up the equation:
12V_cone = V_cylinder
12(πr^3) = 540π
r^3 = 45
Taking the cube root of both sides, we get:
r = 3 cm
Therefore, the radius of the hemisphere is 3 cm, which is option A.
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Community Answer
A solid cylinder of diameter 12 cm and height 15 cm is melted and reca...
Radius of cylinder = 12/2 = 6 cm
height = 15 cm
Volume of cylinder = πr2h
= 22/7 x 62 x 15
= 540π cm3
Volume of 12 toys = 540π cm3
Volume of 1 toy = 540π/12 = 45π cm3
Let the radius of the hemisphere be r cm.
Height of the cone = 3r cm
Volume of one toy = Volume of hemisphere + volume of cone
Now
height = 15 cm
Volume of cylinder = πr2h
= 22/7 x 62 x 15
= 540π cm3
Volume of 12 toys = 540π cm3
Volume of 1 toy = 540π/12 = 45π cm3
Let the radius of the hemisphere be r cm.
Height of the cone = 3r cm
Volume of one toy = Volume of hemisphere + volume of cone
Now
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A solid cylinder of diameter 12 cm and height 15 cm is melted and recast into 12 toys in the shape of a right circular cone mounted on a hemisphere. Find the radius of the hemisphere, if height of the cone is 3 times the radius.a)3 cmb)6 cmc)5 cmd)9 cmCorrect answer is option 'A'. Can you explain this answer?
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