With a radius of 12 (with an error of 0.0001) and a height of 10 (with an error of 0.0015), the error in the volume of the cylinder is approximately 386.21 cubic units.
To find the error in the volume (V) of a cylinder using the formula V = πr³h, considering errors in both radius (Δr) and height (Δh), we can use the formula for error propagation:
Delta V = |(partial V / partial r) * Delta r| + |(partial V / partial h) * Delta h|
Taking the partial derivatives:
partial V / partial r = 3πr²h
partial V / partial h = πr³
Now, substitute the given values: r = 12, h = 10, Δr = 0.0001, and Δh = 0.0015:
Delta V = |3π * 12² * 10 * 0.0001| + |π * 12³ * 0.0015|
Calculate the values:
Delta V = |108π| + |20736π * 0.0015|
Delta V ≈ 339.29 + 46.92
Delta V ≈ 386.21
Therefore, the error in the volume is approximately 386.21 cubic units.
The question probable may be:
How does the error in the volume (V) of a cylinder, calculated using the formula V = πr³h, change when the radius (r) is measured as 12 with an error (Δr = 0.0001) and the height (h) is measured as 10 with an error (Δh = 0.0015)? Provide a step-by-step calculation of the error in V.