JEE PYQ: Units & Measurements - Question ID e02d9edf02d5 (JEE Main 2019)

In measurements involving uncertainties (errors), the volume of a cylinder is calculated using the formula: $V = \pi r^2 h$ where - $r$ is the radius (half the diameter), - $h$ is the height, - $\pi$ is a constant (taken as 3.142 for 4 significant figures).

When quantities with uncertainties are multiplied or raised to powers, the relative error (fractional uncertainty) propagates according to the rules of error propagation. For a product of powers, the relative error in the result is: $\frac{\Delta V}{V} = 2 \frac{\Delta r}{r} + \frac{\Delta h}{h}$ This formula arises because the radius is squared, doubling its relative error contribution.

Step 1: Extract given data and convert diameter to radius
Diameter $D = 12.6 \pm 0.1$ cm
Height $h = 34.2 \pm 0.1$ cm
Radius $r = \frac{D}{2} = \frac{12.6}{2} = 6.3$ cm
Uncertainty in radius $\Delta r = \frac{\Delta D}{2} = \frac{0.1}{2} = 0.05$ cm

Step 2: Compute nominal volume
$V = \pi r^2 h = 3.142 \times (6.3)^2 \times 34.2$ First compute $r^2 = 6.3^2 = 39.69$
Then $V = 3.142 \times 39.69 \times 34.2$
Multiply step-by-step:
$3.142 \times 39.69 = 124.72$ (approx)
$124.72 \times 34.2 = 4265.4$ cm³
Rounding to 4 significant figures (since diameter and height are given to 3 significant figures), $V = 4260$ cm³.

Step 3: Compute relative errors
Relative error in radius $\frac{\Delta r}{r} = \frac{0.05}{6.3} = 0.00794$
Relative error in height $\frac{\Delta h}{h} = \frac{0.1}{34.2} = 0.00292$
Total relative error in volume: $\frac{\Delta V}{V} = 2 \frac{\Delta r}{r} + \frac{\Delta h}{h} = 2 \times 0.00794 + 0.00292 = 0.0188$

Step 4: Compute absolute error in volume
$\Delta V = V \times \frac{\Delta V}{V} = 4260 \times 0.0188 = 79.9$ cm³
Rounding to 1 significant figure (since errors are typically reported to 1 significant figure), $\Delta V = 80$ cm³.

Step 5: Final volume with uncertainty
Volume = $4260 \pm 80$ cm³.

Step 6: Match with given options
Option D: $4260 \pm 80$ cm³ matches our result.

Trap 1: Students often forget to halve the diameter uncertainty when converting to radius. This leads to overestimating $\Delta r$ and consequently $\Delta V$.
Trap 2: Incorrect rounding of the nominal volume. Since diameter and height are given to 3 significant figures, the volume should be reported to 3 or 4 significant figures, not more.
Trap 3: Misapplying error propagation rules. Some students add absolute errors directly, which is incorrect for products or powers.
Exam Tip: Always convert diameter to radius first, compute relative errors, and round the final uncertainty to 1 significant figure. This ensures consistency and avoids over-precision.