The radius of a circular disk is given as 51 cm with a maximum error of 0.04 cm.
Use differentials to estimate the maximum error and percent error in the calculated area of the disk.
To estimate the maximum error and percent error in the calculated area of the disk, we can use differentials. Let's start by finding the formula for the area of a circular disk.
The formula for the area of a circle is given by A = π * r^2, where A is the area and r is the radius.
In our case, the radius of the circular disk is given as 51 cm. To find the maximum error in the radius, we need to consider the maximum error of 0.04 cm.
1. Finding the maximum error:
To estimate the maximum error, we can use differentials. The differential of the area formula is given by dA = 2π * r * dr, where dA is the differential of the area, r is the radius, and dr is the change in the radius.
Since the maximum error in the radius is 0.04 cm, we can substitute this value for dr in the differential equation:
dA = 2π * 51 * 0.04
= 4.064π cm^2
Therefore, the maximum error in the calculated area of the disk is approximately 4.064π cm^2.
2. Finding the percent error:
To find the percent error, we need to calculate the relative error. The relative error is given by:
Relative Error = (Maximum Error / Calculated Value) * 100
The Calculated Value in this case is the actual area of the disk, which can be calculated using the given radius of 51 cm:
Calculated Value = π * (51 cm)^2
= 2601π cm^2
Now, let's substitute the values into the formula to find the percent error:
Percent Error = (4.064π / (2601π)) * 100
≈ 0.156%
Therefore, the estimated maximum error in the calculated area of the disk is approximately 4.064π cm^2, and the percent error is approximately 0.156%.