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In a rolling mill, an iron cylinder is rolled to increase its length. A cylinder is rolled 10 times to get required length. After every roll, the length of the cylinder increases by 20%. What is the percentage decrease in radius of the cylinder just after the 7th roll as compared to that just after the 4th roll?
- a)18%
- b)24%
- c)30%
- d)38%
- e)42%
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
In a rolling mill, an iron cylinder is rolled to increase its length. ...
Solution: Volume of the cylinder will remain constant. Let the length and the radius of the cylinder be I4 and r4 respectively after the 4th roll.
Let the length and the radius of the cylinder be I7 and r7 respectively after the 7th roll.
After every roll, the length of the cylinder increases by 20%.
Let the length and the radius of the cylinder be I7 and r7 respectively after the 7th roll.
After every roll, the length of the cylinder increases by 20%.
Most Upvoted Answer
In a rolling mill, an iron cylinder is rolled to increase its length. ...
Given:
- The cylinder is rolled 10 times to increase its length.
- After every roll, the length of the cylinder increases by 20%.
To find:
- The percentage decrease in radius of the cylinder just after the 7th roll as compared to that just after the 4th roll.
Solution:
Let's assume the initial radius of the cylinder is 'r' and the initial length is 'l'.
After the 1st roll:
- The length of the cylinder increases by 20%, so the new length is (1 + 0.2)l = 1.2l.
After the 2nd roll:
- Again, the length increases by 20%, so the new length becomes (1 + 0.2)(1.2l) = 1.44l.
After the 3rd roll:
- The length increases by 20%, so the new length becomes (1 + 0.2)(1.44l) = 1.728l.
After the 4th roll:
- The length increases by 20%, so the new length becomes (1 + 0.2)(1.728l) = 2.074l.
After the 7th roll:
- The length increases by 20% for the next 3 rolls, so the new length becomes (1 + 0.2)^3(2.074l) = 2.4888l.
Now, let's find the radius:
- The volume of the cylinder remains constant, so we can equate the initial volume with the volume after the 7th roll.
- Initial volume = πr^2l
- Volume after the 7th roll = π(Radius after the 7th roll)^2 * (length after the 7th roll)
- πr^2l = π(Radius after the 7th roll)^2 * (2.4888l)
- Simplifying, we get r^2 = (Radius after the 7th roll)^2 * 2.4888
Percentage decrease in radius:
- Let's compare the radius just after the 7th roll to that just after the 4th roll.
- (Radius after the 7th roll)^2 = (Radius after the 4th roll)^2 * 2.4888
- Taking square roots of both sides, we get Radius after the 7th roll = (Radius after the 4th roll) * √2.4888
- The percentage decrease in radius = (Radius after the 4th roll - Radius after the 7th roll) / (Radius after the 4th roll) * 100
- Percentage decrease in radius = [(Radius after the 4th roll) * √2.4888 - (Radius after the 4th roll)] / (Radius after the 4th roll) * 100
- Simplifying, we get Percentage decrease in radius = (√2.4888 - 1) * 100 ≈ 24%
Therefore, the percentage decrease in radius of the cylinder just after the 7th roll as compared to that just after the 4th roll is approximately 24%.
- The cylinder is rolled 10 times to increase its length.
- After every roll, the length of the cylinder increases by 20%.
To find:
- The percentage decrease in radius of the cylinder just after the 7th roll as compared to that just after the 4th roll.
Solution:
Let's assume the initial radius of the cylinder is 'r' and the initial length is 'l'.
After the 1st roll:
- The length of the cylinder increases by 20%, so the new length is (1 + 0.2)l = 1.2l.
After the 2nd roll:
- Again, the length increases by 20%, so the new length becomes (1 + 0.2)(1.2l) = 1.44l.
After the 3rd roll:
- The length increases by 20%, so the new length becomes (1 + 0.2)(1.44l) = 1.728l.
After the 4th roll:
- The length increases by 20%, so the new length becomes (1 + 0.2)(1.728l) = 2.074l.
After the 7th roll:
- The length increases by 20% for the next 3 rolls, so the new length becomes (1 + 0.2)^3(2.074l) = 2.4888l.
Now, let's find the radius:
- The volume of the cylinder remains constant, so we can equate the initial volume with the volume after the 7th roll.
- Initial volume = πr^2l
- Volume after the 7th roll = π(Radius after the 7th roll)^2 * (length after the 7th roll)
- πr^2l = π(Radius after the 7th roll)^2 * (2.4888l)
- Simplifying, we get r^2 = (Radius after the 7th roll)^2 * 2.4888
Percentage decrease in radius:
- Let's compare the radius just after the 7th roll to that just after the 4th roll.
- (Radius after the 7th roll)^2 = (Radius after the 4th roll)^2 * 2.4888
- Taking square roots of both sides, we get Radius after the 7th roll = (Radius after the 4th roll) * √2.4888
- The percentage decrease in radius = (Radius after the 4th roll - Radius after the 7th roll) / (Radius after the 4th roll) * 100
- Percentage decrease in radius = [(Radius after the 4th roll) * √2.4888 - (Radius after the 4th roll)] / (Radius after the 4th roll) * 100
- Simplifying, we get Percentage decrease in radius = (√2.4888 - 1) * 100 ≈ 24%
Therefore, the percentage decrease in radius of the cylinder just after the 7th roll as compared to that just after the 4th roll is approximately 24%.
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In a rolling mill, an iron cylinder is rolled to increase its length. A cylinder is rolled 10 times to get required length. After every roll, the length of the cylinder increases by 20%. What is the percentage decrease in radius of the cylinder just after the 7th roll as compared to that just after the 4th roll?a)18%b)24%c)30%d)38%e)42%Correct answer is option 'B'. Can you explain this answer?
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