Question

An astronaut using a camera measures the radius of the Earth as 3985 mi with an error of

100 mi. Using differentials, estimate the relative and percentage error of using this radius measurement to calculate the volume of the Earth, assuming the planet is a perfect sphere.
The approximate relative error is:

The approximate percentage error is:
%

Note: enter an answer accurate to 4 decimal places.

Answer 1

The relative error in the volume of the Earth is estimated to be 0.0753, and the approximate percentage error is 7.53% when using the radius measurement of 3985 miles with an error of 100 miles.

Step-by-step explanation:

An astronaut measures the radius of the Earth as 3985 miles with an error of 100 miles. To estimate the relative and percentage error of the Earth's volume calculated using this radius, we can use differentials. If the Earth is a perfect sphere, its volume (V) is given by V = ⅓πr^3, where r is the radius.

We can find the differential of V, dV, which represents the change in volume in response to a small change in radius, dr. The formula for dV is dV = 4πr^2dr. The relative error in volume is ∆V / V, and the percentage error is then calculated by multiplying the relative error by 100%.

By substituting r = 3985 miles and dr = 100 miles, we can estimate the relative and percentage error:

• dV = 4π(3985)^2(100)
• ∆V / V = (dV / V) = (4π(3985)^2(100)) / (⅓π(3985)^3)
• Relative error = 3(100) / 3985 ≈ 0.0753
• Percentage error = 0.0753 × 100% = 7.53%

The approximate relative error is: 0.0753.

The approximate percentage error is: 7.53% %.