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

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

[@ahnafkhan8115]

If the speed of the shadow was hypothetically constant, then the time taken for the owl to descend 15 feet at its constant speed would be the same time for the shadow to travel those last 30 feet?

[@head85]

@@ahnafkhan8115 yes

[@davidsweeney111]

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

[@Ali-bc7rk]

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 :)

[@ccriztoff]

Creepy SOB

[@athenango8713]

This is the most aestheticly related rate problem ive seen!

[@lucymcmmck]

Very helpful. I like the drawings!

[@00chips]

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.

[@commonmancrypto1648]

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. 100000 as the multiplier. (use the A column as headers.) In cell B7 write =((((B3/(B1-(B2+B5)))-(B3/(B1-B2)*B1))+(B3/(B1-B2)*B1)-(B3/(B1-(B2-B5))))*B6)*B4 :-)

[@allygong]

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

[@moseslocke]

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

[@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

[@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. Sweeeeeeeet.

[@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.

[@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.

[@moseslocke]

Thanks for the clarification

[@codjooliviersossa4717]

merci pour l'apport de clarification

[@nikhileshkale2665]

did not get anything. however, toooooo gooood.

[@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.

[@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

[@mikefly320]

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

[@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.

[@pandoraj2113]

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

[@DiaStarvy]

-40 ft / second is the average velocity. -160 ft / second is the instantaneous velocity at that moment.

[@DracoRenaissance]

What if the mouse reacts to the shadow an runs away from it. What if the owl predicts mouse position and adjustes pursuit

[@Rayden440]

You just assumed that the owl travelled at a constant speed (20ft/s) throughout the 15 ft. However, the speed given was the instantaneous speed for the owl as it dives from the height of 15ft. You cannot simply do what you did because the speed of the owl is slowing down as the owl approach the ground.

[@comic4relief]

Great, except isn't speed by definition non-negative?

[@RockbandDrummer321]

speed is moving to the left in the x direction signaling a negative value

[@vivekpanchagnula815]

he meant velocity

[@inventorOz84]

This is a great lecture, I wish I had this channel 10years ago, I might have done better at calculus. Also, if you are wondering a real world scenario example is MOUSE is the target, OWL is the B2 stealth bomber, and the computer has to simulate where the bomb has to be released etc etc.how long will the bomb stay in the air, how much friction etc...

[@Juxtaroberto]

If these extenuating circumstances were in effect, the problem would warn you of them. You make the natural assumption that information not given is irrelevant.

[@DiaStarvy]

The shadow "decelerates" as the owl gets closer to the ground.

[@0Sebek0]

If I were the mouse I'd be confused to see something coming that fast from te side!

[@walid562]

I wish that you started using Meters instead of feet, a more international unit of measurement...

[@kamilkoczurek484]

It's about numbers, not units. :P The important part is that challenges on Khan Academy use meters.

[@cariboux2]

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?

[@fikri4u]

No mouse has been harmed in any way in this calculation..... I hope.

[@moseslocke]

What am doing wrong. If the owl travels at 20 ft / sec. He travels the 15 feet in .75 seconds, and the shadow also travels 30 feet in the same .75 seconds. So why doesn't the shadow move at 40 ft / second?

[@head85]

The shadow does not move at a constant speed like the owl. The shadow moves slower as it approaches the mouse

[@tristanmacgillvray]

Sal, in every video you always draw your "with respect to d" d totally different from your top d. People will think that your doing it for a reason.

[@geetarwanabe]

Thats a very car shaped mouse

[@narcssius]

Is this a homework question?..

[@ashwinvishwakarma2531]

ye

[@blerton14]

why did you take the derivative of the ratios when you could of set them equal to each other (20/(10+x)) = (15/x)... you just cross multiply and solve for x

[@CKJK954T]

blerton pajaziti Most Mistakes with Related Rates occur when students plug values in too early before setting up the proper equation to differentiate. Holding off prevents potential mistakes in the long run.