Use differentials to approximate the maximum error propagated when calculating the volume of a sphere, if the radius is 5 + or - .01 meters
To approximate the maximum error propagated when calculating the volume of a sphere, we can use differentials. The formula for the volume of a sphere is:
V = (4/3) * π * r^3
Let's assume the radius is given as (5 ± 0.01) meters. We can set up the differential equation to find the maximum error propagated in the volume:
dV = (∂V/∂r) * dr
First, let's find the derivative of the volume formula with respect to the radius (∂V/∂r):
∂V/∂r = 4πr^2
Now, we can substitute the given radius and its error:
r = 5 meters, dr = 0.01 meters
Plugging these values into the differential equation, we have:
dV = (4π(5)^2) * (0.01)
Simplifying further:
dV = 100π * 0.01
dV ≈ 3.14
Therefore, the maximum error propagated when calculating the volume of the sphere is approximately 3.14 cubic meters.