Answer:
A) Maximum error = 170.32 cm³
B)Relative error = 0.0575
Step-by-step explanation:
A) Formula for circumference is: C = 2πr
Differentiating with respect to r, we have;
dC/dr = 2π
r is small, so we can write;
ΔC/Δr = 2π
So, Δr = ΔC/2π
We are told that ΔC = 0.5.
Thus; Δr = 0.5/2π = 0.25/π
Now, formula for Volume of a sphere is;
V(r) = (4/3)πr³
Differentiating with respect to r, we have;
dV/dr = 4πr²
Again, r is small, so we can write;
ΔS/Δr = 4πr²
ΔV = 4πr² × Δr
Rewriting, we have;
ΔV = ((2πr)²/π) × Δr
Since C = 2πr, we now have;
ΔV = (C²/π)Δr
ΔV will be maximum when Δr is maximum
Thus, ΔV = (C²/π) × 0.25/π
C = 82 cm
Thus;
ΔV = (82²/π) × 0.25/π
ΔV = 170.32 cm³
B) Formula for relative error = ΔV/V
Relative error = 170.32/((4/3)πr³)
Relative error = 170.32/((4/3)C³/8π³)
Relative errror = 170.32/((4/3)82³/8π³)
Relative error = 170.32/2963.744
Relative error = 0.0575
