If the diameter of a wire is increased by 10%, by how much per cent approximately, will its length be decreased, if the volume remains the same?
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15
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16
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17
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18
Volume of a cylinder V = πr²h = (π/4)d²l. When volume is constant, d²l = constant. If diameter increases by 10%, new diameter = 1.1d, so d² increases by 1.21 times. To keep volume constant, length must decrease by 1/1.21 ≈ 0.826 times, which is a 17.4% decrease, approximately 17%.
To answer this question, let's consider the relationship between the diameter, length, and volume of a wire.
The volume of a wire can be calculated using the formula for the volume of a cylinder:
[ V = \pi r^2 h ]
Where:
- ( V ) is the volume of the wire
- ( r ) is the radius of the wire (half of the diameter)
- ( h ) is the length of the wire
Since the volume remains the same, we can write:
[ \pi r_1^2 h_1 = \pi r_2^2 h_2 ]
Where:
- ( r_1 ) is the initial radius of the wire
- ( r_2 ) is the final radius of the wire (after increasing the diameter by 10%)
- ( h_1 ) is the initial length of the wire
- ( h_2 ) is the final length of the wire
We are given that the diameter of the wire is increased by 10%. Since the radius is half of the diameter, the radius will also increase by 10%.
Let's assume the initial radius is ( r ). Then the final radius will be ( r + 0.1r = 1.1r ).
Substituting these values into the equation for volume, we get:
[ \pi (1.1r)^2 h_2 = \pi r^2 h_1 ]
Simplifying, we find:
[ (1.1)^2 r^2 h_2 = r^2 h_1 ]
Cancelling ( r^2 ) from both sides, we get:
[ (1.1)^2 h_2 = h_1 ]
Dividing both sides by ( h_1 ), we find:
[ (1.1)^2 = \frac{h_1}{h_2} ]
Now, let's calculate the approximate percentage decrease in length:
[ \text{Percentage decrease} = \left(1 - \frac{h_2}{h_1}\right) \times 100 ]
Substituting the value of (\frac{h_1}{h_2}) from the previous equation, we get:
[ \text{Percentage decrease} = \left(1 - (1.1)^2\right) \times 100 ]
Calculating this value, we find:
[ \text{Percentage decrease} \approx 0.17 \times 100 = 17\% ]
Therefore, the approximate percentage by which the length of the wire will decrease is 17%.
Let's go through each option to understand why it is correct or incorrect:
Option A) 15 - This option is incorrect because the approximate percentage decrease in length is not 15%. Option B) 16 - This option is incorrect because the approximate percentage decrease in length is not 16%. Option C) 17 - This option is correct because the approximate percentage decrease in length is approximately 17%. Option D) 18 - This option is incorrect because the approximate percentage decrease in length is not 18%.
The correct answer is C) 17. This option is correct because the approximate percentage decrease in length of the wire is approximately 17%.