Chapter 4 | Applications of Derivatives
363
dV =4 πr 2 dr .
Using the measured radius of 4000 mi, we can estimate
−4 π (4000) 2 (80) ≤ dV ≤4 π (4000) 2 (80).
To estimate the relative error, consider dV
V . Since we do not know the exact value of the volume
V , use the
measured radius r =4000mi to estimate V . Weobtain V ≈ ⎛ ⎝ 4 3 ⎞
⎠ π (4000) 3 . Therefore the relative error satisfies
−4 π (4000) 2 (80) 4 π (4000) 3 /3
π (4000) 2 (80) 4 π (4000) 3 /3 ,
≤ dV
V ≤ 4
which simplifies to
−0.06≤ dV
V ≤0.06.
The relative error is 0.06 and the percentage error is 6%.
4.11 Determine the percentage error if the radius of Earth is measured to be 3950 mi with an error of ±100 mi.
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