The radius of a sphere was measured and found to be 18 cm with a possible error in measurement of at most 0.05 cm. What is the maximum error in using this value of the radius to compute the volume of the sphere?
Question
The radius of a sphere was measured and found to be 18 cm with a possible error in measurement of at most 0.05 cm. What is the maximum error in using this value of the radius to compute the volume of the sphere?
Solution
The volume V of a sphere is given by the formula V = 4/3 * π * r³, where r is the radius of the sphere.
The differential of V with respect to r is given by dV/dr = 4πr².
The maximum error ΔV in the volume is then given by ΔV = |dV/dr * Δr|, where Δr is the maximum error in the radius.
Substituting the given values:
ΔV = |4π * (18 cm)² * 0.05 cm|
= |4 * 3.14159 * 324 cm² * 0.05 cm|
= 203.21274 cm³
So, the maximum error in using this value of the radius to compute the volume of the sphere is approximately 203.21 cubic centimeters.
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