Related rates: shadow | Applications of derivatives | AP Calculus AB | Khan Academy

Sal solves a related rates problem about the shadow an owl casts as it's hunting a mouse. Created by Sal Khan.

AP Calculus AB on Khan Academy: Bill Scott uses Khan Academy to teach AP Calculus at Phillips Academy in Andover, Massachusetts, and heÕs part of the teaching team that helped develop Khan AcademyÕs AP lessons. Phillips Academy was one of the first schools to teach AP nearly 60 years ago.

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[khanacademy]

The speed of the shadow is not constant. Imagine how far the shadow is when the owl is only slightly below 20ft (it is infinitely far when the own is at the same height as the lamp) and how far it "travels" by the time the owl is even at 15 feet.

[Ali-bcrk]

Perfect drawing, why don't you start a drawing course sal? You actually have a unique method of drawing,
I really like to learn it :)

[davidsweeney]

I cant get enough of this, I'm completely addicted, its fantastic !

[allygong]

thanks!! unique example and it was very helpful :)

[commonmancrypto]

Values for Lamp height, Owl Height, Mouse distance from the base of the light, owl speed, approximator, and multiplier in cells B1, and B6. 0.0001 as the approximator. as the multiplier. (use the A column as headers.) In cell B7 write :-)

[athenango]

This is the most aestheticly related rate problem ive seen!

[Deksudo]

I solved it by finding the equation for a line that goes thru the top of the lamp and the owl's position at t0, then found the shadow's distance from the lamp by finding the x-intercept of the line (accepted the ground as y=0) and onwards with implicit differentation from there... That was a lot of work, got the correct result though

[chips]

Sal made this a simplified version of a problem, dx/dt is actually changing, this is just the value of dx/dt at that instant.

[moseslocke]

Thank you for the clarification, and adding to my understanding.

[DracoRenaissance]

Nice. How would you solve a slightly more general problem where the mouse is running in arbitrary continuous path, probably a bezier curve or spline, and the owl is diving following the mouse at say constant speed, a 3D version of a pursuit curve. What would be the velocity (a vector), of the shadow?
That sounds like probability and vector calculus combined.

[nikhileshkale]

did not get anything. however, gooood.

[pandoraj]

how can this video be watched 152k times with only 37 comments?? your videos are helpful for my preparation for uni and GREAT DRAWING!!

[geetarwanabe]

Hmm that is an odd one. Working out the similar triangles myself gets the same result as khan.

Either using similar triangles doesn't take into account the relative motion of the light source being fixed in space or using basic v=d/t doesn't.

Repeating the calcs with y=7.5 gives the answer of dx/dt=24ft/s and so it's likely that using just basic similar triangles doesn't take into account the relative motion

[codjooliviersossa]

merci pour l'apport de clarification

[Twizzzle]

I believe the average speed over the entire distance is 40ft/second however the speed varies over the entire distance. I.e the closer the shadow gets to the mouse the slower it moves. I think this problem is solving "What is the speed of the shadow at that point." Try solving this again the same way Sal did it however use a small value of X to see this.

[DiaStarvy]

It's not as simple as he makes it out to be. The average velocity is -40 ft / second, but the instantaneous speed is -160 ft / second. The shadow "decelerates" as the owl gets closer to the ground.

[mikefly]

these are thinks of these problems? why would an owl fly straight down and not in a swooping motion...anything for a problem

[Skandalos]

Thats correct but it should have been mentioned already in the video. Most people probably dont even know yet at which point of its path the shadows speed is at -160ft/s.

Also, the answer 40ft/s isnt really a wrong one, since this is the average speed of the shadow on its path.

[DiaStarvy]

The bird doesn't decelerate; the shadow does. When the owl is just slightly below the lamp, the shadow is cast very far away. Moving the owl even just a bit lower will move the shadow a lot closer. However, when the owl is very close to the ground, moving the owl by the same amount doesn't make much difference to the shadow's position. Try it out yourself with a lamp if you don't understand.

[cariboux]

I don't see how that relationship works, x/y : (x+10)/20. If that's the case, then when x = 0, the relationship should be 1/2, but it can't be because in the smaller triangle, both x and y will be 0 when x = 0, which is 0/0, not 1/2. What the heck am I missing?