Differentials — Question 4

Question 4

A right circular cylinder has a height of 12 cm, and its radius is measured to be 5 cm with a possible error of 0.1 cm.

• Use differentials to approximate the maximum error in the calculated volume of the cylinder.

• Estimate the relative error and percentage error in the volume.

Question 4 - Solution

The volume of a right circular cylinder is given by: $$V = \pi r^2 h$$

Given: $r = 5 \, \text{cm}, \quad dr = 0.1 \, \text{cm}, \quad h = 12 \, \text{cm}$

(a) Use differentials to approximate the error:

$$dV = \frac{dV}{dr} \cdot dr = 2\pi r h \cdot dr$$ $$dV = 2\pi (5)(12)(0.1) = 12\pi \, \text{cm}^3 \approx \boxed{37.70 \, \text{cm}^3}$$

(b) Estimate the relative error and percentage error:

$$V = \pi r^2 h = \pi (5)^2 (12) = 300\pi \approx 942.48 \, \text{cm}^3$$ $$\text{Relative error} = \frac{dV}{V} = \frac{12\pi}{300\pi} = \frac{12}{300} = 0.04$$ $$\text{Percentage error} = 0.04 \times 100 = \boxed{4\%}$$

Final Answer:

• Maximum error in volume: $\boxed{37.70 \, \text{cm}^3}$

• Relative error: $\boxed{0.04}$

• Percentage error: $\boxed{4\%}$