Differentials — Question 10

Question 10

The radius of a right circular cylinder is measured as $r = 10 \, \text{cm}$ with a possible error of $0.2 \, \text{cm}$, and the height is measured as $h = 20 \, \text{cm}$ with a possible error of $0.3 \, \text{cm}$.

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

• Estimate the percentage error in the volume.

Question 10 - Solution

We are given: $$r = 10 \, \text{cm}, \quad dr = 0.2 \, \text{cm}, \quad h = 20 \, \text{cm}, \quad dh = 0.3 \, \text{cm}$$

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

Use differentials: $$dV = \frac{\partial }{\partial r} dr + \frac{\partial }{\partial h} dh = (2\pi r h) \cdot dr + (\pi r^2) \cdot dh$$

Substitute values: $$dV = 2\pi(10)(20)(0.2) + \pi(10)^2(0.3)$$ $$= 2\pi(200)(0.2) + \pi(100)(0.3) = \pi(80) + \pi(30) = \pi(110)$$

$$\boxed{dV \approx 345.58 \, \text{cm}^3}$$

(a) Estimated maximum error: $\boxed{345.58 \, \text{cm}^3}$

(b) Percentage error:

$$V = \pi r^2 h = \pi(100)(20) = 2000\pi \approx 6283.19 \, \text{cm}^3$$ $$\text{Percentage error} = \frac{345.58}{6283.19} \times 100 \approx \boxed{5.5\%}$$

Final Answers:

• Maximum error in volume: $\boxed{345.58 \, \text{cm}^3}$

• Percentage error: $\boxed{5.5\%}$