Answered: The base radius and height of a right…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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The base radius and height of a right circular cone are measured as 10 cm and 25 cm, respectively,
with a possible error in measurement of as much as 0.1 cm in each. Use differentials to estimate the
ar’h
maximum error in the calculated volume of the cone. ( V =
3

Expert Solution

Step 1

Given formula for the volume of the cone is $V = \frac{π r^{2} h}{3}$

Now, the total differential of $V$ can be expressed as

$\begin{array}{l} d V = V_{r} d r + V_{h} d h \\ \Rightarrow d V = \frac{\partial V}{\partial r} d r + \frac{\partial V}{\partial h} d h \\ \Rightarrow d V = (\frac{\partial}{\partial r} (\frac{π r^{2} h}{3})) d r + (\frac{\partial}{\partial h} (\frac{π r^{2} h}{3})) d h \\ \Rightarrow d V = \frac{2 π r h}{3} d r + \frac{π r^{2}}{3} d h \end{array}$

Now, we know that $r = 10 c m$ and $h = 25 c m$

And there is a possible error in measurement of as much as $0.1 c m$ in each dimension.

This means $d r \approx Δ r = 0.1 c m$

And $d h \approx Δ h = 0.1 c m$

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