The radius r of a solid cylinder is measured as (2.0 \pm 0.1)\text{ cm} and its height h i… — JAMB UTME

The radius $r$ of a solid cylinder is measured as $(2.0 \pm 0.1)\text{ cm}$ and its height $h$ is measured as $(5.0 \pm 0.1)\text{ cm}$ . What is the maximum percentage error in the calculated volume of the cylinder?

A
$7.0\%$
B
$8.0\%$
C
$10.0\%$
$12.0\%$ Cevap

Cevap

12.0\%

The formula for the volume of a cylinder is

V = \pi r^2 h

. According to the principles of error propagation, the fractional error in

V

\frac{\Delta V}{V} = 2\frac{\Delta r}{r} + \frac{\Delta h}{h}

. Substituting the values:

\frac{\Delta r}{r} = \frac{0.1}{2.0} = 0.05

(

5.0\%

) and

\frac{\Delta h}{h} = \frac{0.1}{5.0} = 0.02

(

2.0\%

). Thus, the maximum percentage error is

2(5.0\%) + 2.0\% = 12.0\%

Adım Adım Çözüm

Calculate the percentage error in the radius measurement

\text{Percentage error in } r = \left(\frac{0.1}{2.0}\right) \times 100\% = 5.0\%

Percentage error is given by the ratio of absolute uncertainty to the measured value multiplied by 100.

Calculate the percentage error in the height measurement

\text{Percentage error in } h = \left(\frac{0.1}{5.0}\right) \times 100\% = 2.0\%

Percentage error of a single linear measurement.

Apply the error propagation formula for the volume of a cylinder

\frac{\Delta V}{V} \times 100\% = 2\left(\frac{\Delta r}{r} \times 100\%\right) + \left(\frac{\Delta h}{h} \times 100\%\right)

Since

V = \pi r^2 h

, fractional errors add up, with the power of any variable acting as a multiplier for its fractional error.

Compute the total maximum percentage error in volume

\text{Percentage error in } V = 2(5.0\%) + 2.0\% = 10.0\% + 2.0\% = 12.0\%

Adding the individual fractional error contributions gives the maximum percentage error.

Anahtar Kavram

Error propagation in derived quantities with exponent powers

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