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A spherical shell of lead, whose external diameter is 18 cm, is melted and recast into a right circular cylinder, whose height is 8 cm and diameter 12 cm. The internal diameter of the shell is:
- a)6√19 cm
- b)619 cm
- c)6(19)1/4 cm
- d)6(19)1/3 cm
Correct answer is option 'D'. Can you explain this answer?
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A spherical shell of lead, whose external diameter is 18 cm, is melted...
To solve this problem, let's break it down into smaller steps.
Step 1: Find the volume of the spherical shell.
The external diameter of the shell is given as 18 cm. Therefore, the radius of the shell is half of the diameter, which is 9 cm. The internal diameter of the shell is not given, so let's assume it as 'd' cm. Therefore, the internal radius of the shell is half of the internal diameter, which is d/2 cm.
The volume of the spherical shell is given by the formula:
Volume = (4/3)π(R^3 - r^3)
where R is the external radius and r is the internal radius.
Substituting the values, we get:
Volume = (4/3)π((9)^3 - (d/2)^3)
= (4/3)π(729 - (d^3/8))
= (4/3)π(729 - d^3/8)
Step 2: Find the volume of the cylinder.
The height of the cylinder is given as 8 cm and the diameter is 12 cm. Therefore, the radius of the cylinder is half of the diameter, which is 6 cm.
The volume of the cylinder is given by the formula:
Volume = πr^2h
where r is the radius and h is the height.
Substituting the values, we get:
Volume = π(6^2)(8)
= 288π
Step 3: Equate the volumes of the shell and the cylinder.
Since the lead is melted and recast into a cylinder, the volume of the shell should be equal to the volume of the cylinder.
(4/3)π(729 - d^3/8) = 288π
Canceling out π and multiplying both sides by 3/4, we get:
729 - d^3/8 = 216
Multiplying both sides by 8, we get:
5832 - d^3 = 1728
Subtracting 5832 from both sides, we get:
-d^3 = -4104
Dividing by -1, we get:
d^3 = 4104
Taking the cube root of both sides, we get:
d = ∛4104
Simplifying, we get:
d ≈ 15.99 cm
Therefore, the internal diameter of the shell is approximately 15.99 cm, which can be rounded off to 16 cm.
Since none of the options provided match the calculated value, it seems that there may be a mistake in the given answer options.
Step 1: Find the volume of the spherical shell.
The external diameter of the shell is given as 18 cm. Therefore, the radius of the shell is half of the diameter, which is 9 cm. The internal diameter of the shell is not given, so let's assume it as 'd' cm. Therefore, the internal radius of the shell is half of the internal diameter, which is d/2 cm.
The volume of the spherical shell is given by the formula:
Volume = (4/3)π(R^3 - r^3)
where R is the external radius and r is the internal radius.
Substituting the values, we get:
Volume = (4/3)π((9)^3 - (d/2)^3)
= (4/3)π(729 - (d^3/8))
= (4/3)π(729 - d^3/8)
Step 2: Find the volume of the cylinder.
The height of the cylinder is given as 8 cm and the diameter is 12 cm. Therefore, the radius of the cylinder is half of the diameter, which is 6 cm.
The volume of the cylinder is given by the formula:
Volume = πr^2h
where r is the radius and h is the height.
Substituting the values, we get:
Volume = π(6^2)(8)
= 288π
Step 3: Equate the volumes of the shell and the cylinder.
Since the lead is melted and recast into a cylinder, the volume of the shell should be equal to the volume of the cylinder.
(4/3)π(729 - d^3/8) = 288π
Canceling out π and multiplying both sides by 3/4, we get:
729 - d^3/8 = 216
Multiplying both sides by 8, we get:
5832 - d^3 = 1728
Subtracting 5832 from both sides, we get:
-d^3 = -4104
Dividing by -1, we get:
d^3 = 4104
Taking the cube root of both sides, we get:
d = ∛4104
Simplifying, we get:
d ≈ 15.99 cm
Therefore, the internal diameter of the shell is approximately 15.99 cm, which can be rounded off to 16 cm.
Since none of the options provided match the calculated value, it seems that there may be a mistake in the given answer options.
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A spherical shell of lead, whose external diameter is 18 cm, is melted and recast into a right circular cylinder, whose height is 8 cm and diameter 12 cm. The internal diameter of the shell is:a)619 cmb)619 cmc)6(19)1/4cmd)6(19)1/3cmCorrect answer is option 'D'. Can you explain this answer?
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A spherical shell of lead, whose external diameter is 18 cm, is melted and recast into a right circular cylinder, whose height is 8 cm and diameter 12 cm. The internal diameter of the shell is:a)619 cmb)619 cmc)6(19)1/4cmd)6(19)1/3cmCorrect answer is option 'D'. Can you explain this answer? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about A spherical shell of lead, whose external diameter is 18 cm, is melted and recast into a right circular cylinder, whose height is 8 cm and diameter 12 cm. The internal diameter of the shell is:a)619 cmb)619 cmc)6(19)1/4cmd)6(19)1/3cmCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A spherical shell of lead, whose external diameter is 18 cm, is melted and recast into a right circular cylinder, whose height is 8 cm and diameter 12 cm. The internal diameter of the shell is:a)619 cmb)619 cmc)6(19)1/4cmd)6(19)1/3cmCorrect answer is option 'D'. Can you explain this answer?.
A spherical shell of lead, whose external diameter is 18 cm, is melted and recast into a right circular cylinder, whose height is 8 cm and diameter 12 cm. The internal diameter of the shell is:a)619 cmb)619 cmc)6(19)1/4cmd)6(19)1/3cmCorrect answer is option 'D'. Can you explain this answer? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about A spherical shell of lead, whose external diameter is 18 cm, is melted and recast into a right circular cylinder, whose height is 8 cm and diameter 12 cm. The internal diameter of the shell is:a)619 cmb)619 cmc)6(19)1/4cmd)6(19)1/3cmCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A spherical shell of lead, whose external diameter is 18 cm, is melted and recast into a right circular cylinder, whose height is 8 cm and diameter 12 cm. The internal diameter of the shell is:a)619 cmb)619 cmc)6(19)1/4cmd)6(19)1/3cmCorrect answer is option 'D'. Can you explain this answer?.
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