Question

The error in the measurement of length and radius of a cylinder are 1% and 2% respectively. The maximum percentage of error in its volume is:

A. 0.03 B. 0.05 C. 0.06 D. 0.1

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Answer

To determine the maximum percentage error in the volume of a cylinder given the errors in measurement, we'll follow these steps:

  1. Formula for the volume of a cylinder: The volume ( V ) of a cylinder is given by: $$ V = \pi r^2 h, $$ where ( r ) is the radius and ( h ) is the height.

  2. Relating percentage errors: The percentage error in the volume ( V ) can be expressed in terms of the percentage errors in radius ( r ) and height ( h ). The percentage error in volume is calculated as: $$ \frac{\Delta V}{V} \times 100. $$

  3. Using error propagation: The relationship for the percentage error in volume ( V ) derived from the product of ( r^2 ) and ( h ) can be approximated as: $$ \frac{\Delta V}{V} \times 100 \approx 2 \left(\frac{\Delta r}{r} \times 100\right) + \left(\frac{\Delta h}{h} \times 100\right). $$

  4. Given errors: The error in the measurement of length (height) of the cylinder is ( 1% ) and the error in the radius is ( 2% ).

  5. Substitute the values: [ \frac{\Delta V}{V} \times 100 = 2 \times 2% + 1% = 4% + 1% = 5%. ]

  6. Convert percentage to a decimal: The fractional error is: $$ \frac{\Delta V}{V} = \frac{5}{100} = 0.05. $$

Therefore, the maximum percentage of error in the volume of the cylinder is $\boxed{5%}$, which corresponds to decimal 0.05.

Answer:

B. 0.05


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