Question

The volume V of a right-circular cylinder is computed using the values d = 3.9 m for the diameter and h = 6.9 m for the height. Use the Linear Approximation to estimate the maximum error R in V if each of these values has a possible error of at most 5%.

Answer 1

V≈82.43

Step-by-step explanation:

The formula to solve for the volume a right-circular cylinder is
`V=\pi r^(2)h`.

The radius is just the diameter divided by 2. So the radius would 1.95.

So now we have to plug in the height and radius to the problem.
`V=\pi 1.95^(2) 6.9`

Then you multiply 1.95 by itself which would be:
`V=\pi 3.8025` times 6.9

`V=\pi 26.23725`

Then you multiply by pi.

V≈82.43

There you have it, hopefully this helped. :)

Answer 2

The maximum error in the volume of a right-circular cylinder with a 5% possible error in measurement of diameter and height can be estimated using Linear Approximation. The total maximum relative error for the volume is 15%, accounting for both the errors in the radius (doubled due to squaring) and height. The maximum error is then found by multiplying this percentage with the calculated volume.

Step-by-step explanation:

The volume V of a right-circular cylinder can indeed be calculated using the formula V = πr²h, where π is Pi, r is the radius of the cylinder, and h is the height of the cylinder. Given that the diameter d is 3.9 m, the radius r will be half of that, which is 1.95 m. The height h provided is 6.9 m. Using these measurements, the exact volume of the cylinder is calculated as:

V = π(1.95 m)²(6.9 m).

When it comes to estimating the maximum error R in V if there is a possible error of at most 5% in measuring the diameter and height, we utilize the concept of Linear Approximation in calculus. The maximum relative error in the volume can be found by adding the relative errors of the dimensions since the volume is directly proportional to each dimension (i.e., the radius and height).

• Maximum relative error in diameter (radius): 5%
• Maximum relative error in height: 5%

Since the volume formula (V = πr²h) involves the radius squared, the relative error in the radius will be doubled when calculating the total relative error for the volume. Therefore, if the maximum relative error in the radius is 5%, the contribution to the volume error from the radius will be 10%. Adding the height's error, the total maximum relative error for the volume would be 15%.

Now the maximum error R can be estimated by multiplying the total maximum relative error by the calculated volume:

R = 0.15 × V (calculated volume).