Question:
If the height of the tower used for L.D.S is increased by 21% then percentage change in range is?
If the height of the tower used for L.D.S is increased by 21% then percentage change in range is?
Updated On: Apr 22, 2024
- 10%
- 21%
- 19%
- 42%
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The Correct Option is A
Approach Solution - 1
The range of the L.D.S. (Laser Distance Sensor) is given by:
Range ∝ \(\sqrt{Height}\)
This means that the range of the L.D.S. is directly proportional to the square root of the height of the tower.
If the height of the tower is increased by 21%, then the new height is:
New Height = Old Height + 0.21 x Old Height = 1.21 x Old Height
So the percentage change in height is 21%.
Now, using the formula above, we can write:
New Range ∝ \(\sqrt{New Height}\)
New Range ∝ \(\sqrt{1.21 x Old Height}\)
New Range ∝ \(\sqrt{1.21} x \sqrt{Old Height}\)
New Range ∝ 1.1 x \(\sqrt{Old Height}\)
Therefore, the new range is 1.1 times the square root of the old range, which is an increase of:
Percentage change in range = (New Range - Old Range) / Old Range x 100%
Percentage change in range = [(1.1 x \(\sqrt{Old Height}\)) - \(\sqrt{Old Height}\)] / \(\sqrt{Old Height}\) x 100%
Percentage change in range = (0.1 / \(\sqrt{Old Height}\)) x 100%
Substituting the value of the percentage change in height, we get:
Percentage change in range = (\(\frac{0.1}{\sqrt{Old Height}}\)) x 100% = (\(\frac{0.1}{\sqrt{\frac{1}{1.21}}}\)) x 100%
Percentage change in range = 10%
Therefore, the percentage change in range is 10%.
Answer. A
Range ∝ \(\sqrt{Height}\)
This means that the range of the L.D.S. is directly proportional to the square root of the height of the tower.
If the height of the tower is increased by 21%, then the new height is:
New Height = Old Height + 0.21 x Old Height = 1.21 x Old Height
So the percentage change in height is 21%.
Now, using the formula above, we can write:
New Range ∝ \(\sqrt{New Height}\)
New Range ∝ \(\sqrt{1.21 x Old Height}\)
New Range ∝ \(\sqrt{1.21} x \sqrt{Old Height}\)
New Range ∝ 1.1 x \(\sqrt{Old Height}\)
Therefore, the new range is 1.1 times the square root of the old range, which is an increase of:
Percentage change in range = (New Range - Old Range) / Old Range x 100%
Percentage change in range = [(1.1 x \(\sqrt{Old Height}\)) - \(\sqrt{Old Height}\)] / \(\sqrt{Old Height}\) x 100%
Percentage change in range = (0.1 / \(\sqrt{Old Height}\)) x 100%
Substituting the value of the percentage change in height, we get:
Percentage change in range = (\(\frac{0.1}{\sqrt{Old Height}}\)) x 100% = (\(\frac{0.1}{\sqrt{\frac{1}{1.21}}}\)) x 100%
Percentage change in range = 10%
Therefore, the percentage change in range is 10%.
Answer. A
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