The volume V of a cylinder is computed using the values 9.8 \text{m} for the diameter and 3 \text{m} for the height. Use the linear approximation to estimate the maximum error in V if each of these values has a possible error of at most 8 \%.

To estimate the maximum error in the volume of the cylinder, we can use the linear approximation.

The formula for the volume of a cylinder is V = πr^2h, where r is the radius and h is the height.

Given that the diameter is 9.8 m, we can find the radius by dividing the diameter by 2:
r = 9.8 m / 2 = 4.9 m

Now, let's calculate the volume using these values:
V = π(4.9 m)^2 * 3 m
V ≈ 225.06π m^3

To find the maximum error in the volume, we need to consider the possible errors in both the diameter and the height.

For the diameter, the possible error is 8% of 9.8 m:
diameter_error = 0.08 * 9.8 m = 0.784 m

Similarly, for the height, the possible error is 8% of 3 m:
height_error = 0.08 * 3 m = 0.24 m

To estimate the maximum error in the volume, we can use the linear approximation by considering the changes in the volume due to the errors in the diameter and height.

Let ΔV be the change in volume due to the errors. Then:
ΔV ≈ (∂V/∂r) * diameter_error + (∂V/∂h) * height_error

The partial derivatives of V with respect to r and h are:
(∂V/∂r) = 2πrh
(∂V/∂h) = πr^2

Using these values, we can calculate the maximum error in V:
ΔV ≈ 2π(4.9 m)(3 m) * 0.784 m + π(4.9 m)^2 * 0.24 m
≈ 476.38π m^3

Therefore, the maximum error in V is approximately 476.38π m^3.

To estimate the maximum error in the volume of the cylinder, we can use the linear approximation. The general formula for the volume of a cylinder is given by:

V = πr^2h

Where r is the radius and h is the height of the cylinder. Since the diameter is given as 9.8 m, we can calculate the radius as follows:

r = d/2 = 9.8/2 = 4.9 m

Now, let's calculate the volume using the given values:

V = π(4.9)^2(3) = 234.915π m^3

To estimate the maximum error, we need to consider the possible error in the diameter and height. The given error is 8% for both values, which means each value may vary by a maximum of 8%.

For the diameter:
Possible error = 8% of 9.8 = 0.08 * 9.8 = 0.784 m

For the height:
Possible error = 8% of 3 = 0.08 * 3 = 0.24 m

To estimate the maximum error in the volume, we need to compute the partial derivatives of V with respect to r and h:

∂V/∂r = 2πrh
∂V/∂h = πr^2

Now, let's calculate the maximum error using the linear approximation formula:

ΔV = (∂V/∂r) * Δr + (∂V/∂h) * Δh

Where Δr and Δh are the possible errors in the radius and height, respectively.

ΔV = (2π(4.9)(3) * 0.784) + (π(4.9)^2 * 0.24)
= (29.952π) + (59.754π)
= 89.706π

Therefore, the estimated maximum error in the volume of the cylinder is 89.706π m^3, where π is approximately 3.14.

Note: It's important to keep in mind that the error calculated here represents an estimation based on the linear approximation. The actual error may differ, especially if the errors in the diameter and height are not independent or have a non-linear relationship with the volume.