SUNY Geneseo Department of Mathematics

Related Rates Examples

Thursday, October 22

Math 221 02
Fall 2020
Prof. Doug Baldwin

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Previous Lecture

Anything You Want to Talk About?

When constants appear as the argument to some function, the whole function result is constant, right? Yes, for example sin(10˚) is a constant.

Misc

SI

The next session will be Sunday 6:00 - 7:30. It will review this week, and talk about problem set questions if time permits.

Mid-Semester Feedback

One theme in feedback surveys from this course and others was students asking for video recordings of classes, or classes that students could attend via Zoom when they can’t come in person.

Unfortunately, there are lots of technical and possibly legal/ethical challenges in doing that routinely. I might be able to make short focused videos outside of class for some things (e.g., Mathematica demonstrations where interaction is important) though.

A related question came up yesterday: could we hold classes via Zoom after Thanksgiving, when everything is online? My experience with such classes isn’t good — there’s just enough distance between people to cut way down on interaction, technical difficulties can shut some people out, and it’s very hard for me to judge how well students are understanding something. But if you strongly favor Zoom classes for a short period, I might be persuaded. Please let me know if you have a preference either for or against.

Related Rates Examples

Based on “Examples of the Process” in section 4.1 and the related rates discussion.

Pancake

This has a good solution in the discussion.

Drone Camera

This is also solved in the discussion, but we reviewed the solution in class.

A drone aircraft carrying a camera is flying at a height of 100 feet, and rising at a rate of 2 feet/minute. The camera takes square photos with a 20 degree field of view, i.e., the square that the camera “sees” extends ± 10 degrees to each side of the line of sight. How fast is the area in the photo changing?

Drone with camera flying over a square patch of ground

Having this picture of the situation, the next step is to write equations relating the relevant quantities. The key to forming those equations is the triangle formed by a vertical line from the camera to the ground and then to the side of the area in the photo:

Line from camera to ground and line from there to edge of square make a right triangle

The angle at the drone inside this triangle is half the field of view, or 10 degrees. The side of the triangle opposite this angle covers half the width of the photo area, and the side adjacent to this angle is the height of the drone, so we can use trigonometry to relate the side of the photo region, and thus its area, to the drone’s height:

L over 2 over h equals tangent of 10 degrees, so A equals 4 h squared times tangent squared of 10 degrees

Now differentiate both sides of this equation to find dA/dt:

d A d T equals 8  h times d h d t times tangent squared of 10 degrees

Finally, plug in numbers we know for height and its rate of change to get the rate of change of the area:

d A d t equals 1600 times tangent squared of 10 degrees

Beware that we couldn’t plug these numbers in before we took derivatives, because otherwise both sides of the equation would have constant values, and thus derivatives of 0. This is a common pitfall in related rates problems.

Spy

Super-spy Jane Bond is hiding under the wall outside Dr. Evil’s compound. Dr. Evil has cameras on telescoping poles that look over the wall for people doing exactly this. But there is a “safe” region right below the wall that the cameras can’t see, with the width of that region, s, depending on how high the camera is. If the wall is 4 meters high, and the nearest camera is on a pole that is currently 5 meters high and extending at a rate of 2 meters per second, how fast is the width of the safe region shrinking?

Spy on one side of wall, telescoping pole with camera on other

The key to connecting the height of the pole (whose rate of change we know) to the width of the safe region (whose rate of change we want to know) is similar triangles. For example, the triangle whose sides are the wall and the safe region is similar to the triangle whose sides are the pole and the safe region plus the distance from the wall to the pole:

Ground, wall, and pole form similar triangles

One of the properties of similar triangles is that ratios of corresponding sides are the same in a pair of similar triangles, so we can use the ratio of the height to the base in both triangles to find an equation relating s (the width of the safe region) to h (the height of the pole):

4 over s equals h over s plus 2, so h over h minus 4 equals s

Now differentiating both sides of this, using the quotient rule on the left, gives ds/dt:

d s d t equals minus 8 over h minus 4 squared times d h d t

And finally, plug in numbers we know in order to find that the safe region is shrinking pretty quickly:

d s d t equals minus 16

Next

More related rates examples.

No new reading, but keep talking about and/or solving example problems in the related rates discussion.

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