# Related Rates The concept of related rates is pretty For example, if a balloon is being filled with air, both the radius $(r)$ and volume $(V)$ of the balloon are increasing over time $(t).$ > [!hint] Strategy: Solving a Related-Rates Problem > The following procedure can be followed when solving related-rates problems > > 1. Assign symbols to all variables involved in the problem. Draw a figure if applicable. > 2. State, in terms of variables, the information that is given and the rate that is under investigation. > 3. Find an equation relating the variables introduced in step 1. > 4. Using the chain rule and [[Implicit Differentiation]], differentiate both sides of the equation found in step 3 with respect to the independent variable. This new equation will relate the derivatives. > 5. Substitute all known values into the equation from step 4, then solve for the unknown rate of change. --- > [!question]+ Practice: Related Rates with Volume of Cones > Water is leaking out of an inverted conical tank at a rate of *8700 cubic centimeters per min* at the same time that water is being pumped into the tank at a constant rate. The tank has a height of *9 meters* and the diameter at the top is *6 meters*. if the water level is rising at a rate of *20 centimeters per minute* when the height of the water is *2.5 meters*, find the rate at which water is being pumped into the tank in cubic centimeters. > > [!hint]+ Hints > > - This problem provides values with different units (meters and centimeters), make sure that you are **properly converting units** them when solving the problem. > > > > - The solution to this problem is not immediately obvious and is an example of how difficult it can be to apply mathematics to real-world scenarios. > > > > - It's important to reevaluate your initial approach to a problem whenever you get stuck, it's very possible that part of the problem is misleading (in this case the difference in volume ($\frac{dV}{dt}$) can be easily confused) > > > [!check]+ Answer > > **Answer:** $445032.305 \frac{cm^3}{min}$ > > --- > > For this equation we will need the volume equation of a cone: $V = \pi r^2 \frac{h}{3}$ > > > > 1. We begin by finding the important values provided by the problem and providing them with an appropriate symbol: > > - $h = 9$ meters > > - $d = 6$ meters, or $r = 3$ meters > > - $\frac{dh}{dt} = 20$ cm per minute > > - $\frac{dV}{dt} = 20$ cubic cm per minute > > i > > --- > [!question]+ Practice: Related Rates with the Pythagorean Theorem > At noon, ship A is *40 nautical miles* due west of ship B, ship A is sailing west at *16 knots* and ship B is sailing north at *22 knots*. How fast (in knots) is the distance between the ships changing at *3 PM*? (Note: 1 knot is a speed of 1 nautical mile per hour) > > > [!hint]- Hints > > (note: 1 knot is a speed of 1 nautical mile per hour) > > > > [Example Solution](https://www.youtube.com/watch?v=U76zi9Jf73A) > > > [!check]- Answer > > aaaa https://www.wyzant.com/resources/answers/839248/water-is-leaking-out-of-an-inverted-conical-tank-at-a-rate-of-10-500-cm3-m https://www.wyzant.com/resources/answers/793030/when-air-expands-adiabatically-without-gaining-or-losing-heat-its-pressure-