Compute the absolute and relative errors in using x to approximate x.
x=pi; c=3.18
3.18-π = 0.0384
0.0384/π = .0122 = 1.22%
A light in a lighthouse 5 kilometers offshore from a straight shoreline is rotating at 4 revolutions per minute. How fast is the beam moving along the shoreline when it passes the point 5 kilometers from the point opposite the lighthouse?
To compute the absolute and relative errors in using x to approximate x, we need to determine the difference between the actual value of x (pi) and the approximate value of x (c).
Absolute Error:
The absolute error represents the difference between the actual and approximate values without considering the scale. It is calculated as the absolute value of the difference between the actual and approximate values.
In this case, the absolute error can be calculated as |x - c|:
Absolute error = |pi - 3.18|
Relative Error:
The relative error takes into account the scale of the values and is calculated as the ratio of the absolute error to the actual value. It provides a more meaningful measurement when comparing errors for different values.
In this case, the relative error can be calculated as |x - c| / |x|, where |x| denotes the absolute value of x:
Relative error = |pi - 3.18| / |pi|
Now, let's compute the absolute and relative errors:
Absolute error = |pi - 3.18| = 0.0384 (rounded to four decimal places)
Relative error = |pi - 3.18| / |pi| ≈ 0.0122 (rounded to four decimal places)
Therefore, the absolute error is approximately 0.0384, and the relative error is approximately 0.0122.