# Related Rate Problems I

Related rate problems provide an early opportunity for students to use calculus in a, more or less, real context and practice implicit differentiation.

One of the problems students have with these problems is that almost all of them involve writing the model or starting equation based on some geometric situation. Students have to switch from calculus to geometry and then back again. When starting out, one of the ways to avoid this is to give a few problems that do not involve any geometry. Once they have the idea of relating the rates by using the derivative, then they may be ready to tackle the geometry.

Here are two examples of related rate problems without geometry (answers at the end).

1. The kinetic energy, K in joules, of a moving object is given by the equation $K=\tfrac{1}{2}m{{v}^{2}}$ where m is the mass of the object and v is its velocity. The mass of a rocket decreases at a constant rate of 25 kg/sec due to the burning of its fuel. When the mass of the rocket is 6000 kg, the velocity is 12 m/sec and increasing at the rate of 2 m/sec/sec. At this instant how fast is the kinetic energy in changing? (The units are joules / sec.)

2. The force, F in Newtons, of a moving object is given by the equation $F=ma$ where m is the mass of the object and a is its acceleration. A rocket sled is propelled along a track with an acceleration given by $a(t)=5{{t}^{2}}+6t\text{ for }t\ge 0$. When t = 6 sec. its mass is 10 kg and is decreasing at the rate of 0.2 kg/sec due to the burning of its fuel. At this instant how fast is the force changing?

The next post will be two out of the ordinary related rate problems (with geometry).

Answers:  1. 142,200 joules / sec.  2.   616.8 Newtons / sec

## 3 thoughts on “Related Rate Problems I”

1. Kate Baker says:

In Problem 2, should the equation for kinetic energy be replaced by F=ma? (I’m also a big fan! Thank you for your wonderful suggestions!)

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• Yes. I need a good proofreader sometimes – often. I made the change. Thanks.

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2. Liz Kearns says:

This is a really great idea! Lin this blog is great!! Sometimes we teachers forget that we need to continually look at the way we are teaching our students. Your blog offers such a variety of ways to look at and to practice different topics. I went to one of your workshops years ago at Drew University when I just began teaching AP it was very helpful!

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