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Related Rates

This note along with the problem set is based on basic applications of differentiation to real-world problems.To understand this note you need to have basic ideas of Differential Calculus.

This note is about problems on related rates-which is the study of change of one variable with respect to another provided that we find a constant parameter providing a relationship between those two variables.AN important point for approaching problems like these is to find that constant variable providing the link between the variables we want to evaluate.

\(\textbf{Problem.}\)Consider a conical tank. Its radius at the top is 5 feet, and it’s 12 feet high. It’s being filled with water at the rate of 3 cubic feet per minute. How fast is the water level rising when it is 7 feet high? (See the given figure)


From the second figure by triangle similarity we have \[\frac{r}{5} = \frac{h}{12}\] \[r = 5 \frac{h}{12}\]

Now we need to find the constant parameter.Clearly it is the rate of change of volume!!Since given \( \dfrac{\text{d}V}{\text{d}t} = 2 \). We have \[V = \frac{1}{3}\pi r^{2} h\]

We put the value of \(r\) in terms of \(h\) to get

\[V = \frac{1}{3}\pi (5 \frac{h}{12})^{2} h\]

\[V = \frac{1}{3}\pi (5 \frac{h}{12})^{2} h\]

\[V = \frac{1}{3}\pi (25 \frac{h^3}{144}) \]

Now we differentiate both sides to get:

\[\dfrac{\text{d}V}{\text{d}t} = \frac{1}{3}\pi (25*3 \frac{h^2}{144})\dfrac{\text{d}h}{\text{d}t} \]

.Now we simply put the values of \(h\) and \(\dfrac{\text{d}V}{\text{d}t}\) to get the correct answer.

Now that you have learnt how to approach these situations try your hand at the problems.They are not that hard......After that you'll see derivatives everywhere.

Note by Eddie The Head
3 years ago

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The problem says that the rate of filling is 3 cubic feet per minute, so why is \(\frac{dv}{dt} = 2\)? Shendy Marcello Yuniar · 3 years ago

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@Shendy Marcello Yuniar Oops...typo....Sorry for that.... Eddie The Head · 3 years ago

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khanacademy.com also has very nice explanations of this. Daniel Liu · 3 years ago

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Nice explanation thanks Kahsay Merkeb · 3 years ago

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