Related Rates - Conical Tank, Ladder Angle & Shadow Problem, Circle & Sphere - Calculus

This calculus video tutorial explains how to solve related rates problems using derivatives. It shows you how to calculate the rate of change with respect to radius, height, surface area, or volume of a sphere, circle, cone, etc. This video contains plenty of examples and practice problems such as the inverted conical tank problem, the ladder angle problem, similar triangle shadow problem, problems with circles, spheres, cubes, cones, squares, and triangles and so forth.
Introduction to Limits:
• Calculus 1 - Introduct...
Derivatives - Fast Review:
• Calculus 1 - Derivatives
Introduction to Related Rates:
• Introduction to Relate...
Derivative Notations:
• dy/dx, d/dx, and dy/dt...
Related Rates - The Cube:
• Related Rate Problems ...
__________________________________
Inflated Balloon & Melting Snowball:
• Related Rates - Inflat...
Gravel Dumped Into Conical Tank:
• Related Rates - Gravel...
Related Rates - Area of a Triangle:
• Related Rates - Area o...
Related Rates - The Ladder Problem:
• Related Rates - The La...
Related Rates - The Distance Problem:
• Related Rates - Distan...
____________________________________
Related Rates - Airplane Problems:
• Related Rates - Airpla...
Related Rates - The Shadow Problem:
• Related Rates - The Sh...
Related Rates - The Baseball Diamond Problem:
• Related Rates - The Ba...
Related Rates - The Angle of Elevation Problem:
• Related Rates - Angle ...
Related Rates - More Practice Problems:
• Related Rates - Conica...
_________________________________
Final Exams and Video Playlists:
www.video-tuto...
Full-Length Videos and Worksheets:
/ collections
Here is a list of problems.
1. Finding dx/dt, dy/dt and dz/dt - Pythagorean Theorem - Right Triangle Trigonometry
2. How to find the rate of change of the distance between the origin and a moving point on the graph if the y-coordinate is increasing
3. The radius of a circle is decreasing at a rate of 4cm/min. How fast is the area and circumference changing when the radius is 8cm?
4. The surface area of a snowball at a rate of 6 square feet per hour, how fast is the diameter changing when the radius is 2 ft?
5. The side length of a square increases at a rate of 3 inches per second, how fast is the area and perimeter of the square changing when the side length is 5 in ?
6. A spherical balloon is inflated with gas at a rate of 900 cubic centimeters per minute (cm^3/min), how fast is the radius of the balloon changing?
7. The side lengths of a cube are increasing at a rate of 5 cm/s, How fast is the surface area and volume increasing?
8. A 13 foot ladder leans against a house. The ladder slides down the wall at a rate of 3 ft/min. How fast is the ladder moving away from the base of the wall when the foot of the ladder is currently 5ft from the wall? How fast is the area of the triangle changing? How fast is the angle between the ground and ladder changing?
9. Gravel is being dumped from a conveyor belt at a rate of 100 cubic feet per min (ft^3/min) forming a conical pile whose base diameter is two times the altitude. How fast is the height changing?
10. Water is leaking out of an inverted conical tank at 500 cm^3/min. The tank has a height of 24 cm and a radius of 6cm. Find the rate at which water is being poured into the tank if the water level is rising at 15cm/min.
11. A street light is mounted on a pole 24 ft tall. A man 6ft tall walks away from the pole at a rate of 4ft/s. How fast is the tip of his shadow moving when he is 20ft from the pole? How fast is the length of his shadow changing at this instant?
12. A spotlight shines on a wall 18m away. If a 2m tall man walks toward the building at a speed of 2m/s, how fast is the length of his shadow on the building changing?
13. Two cars are moving starting from the same point.
14. At 1:00pm, ship B is 150 miles from ship A. Ship A is moving 30mph north and ship B is moving 20mph south. How fast is the distance changing at 3:00pm?
15. Airplane Problem - Travels Horizontally at an altitude of 3 miles. Radar Station Below.
16. Airplane Observer Problem - Rate of Change of Angle of Elevation
17. Baseball Diamond Square Problem - Speed in ft/s.
18. Water trough problem

[@TheOrganicChemistryTutor]

Final Exams and Video Playlists: www.video-tutor.net/ Full-Length Videos & Worksheets: www.patreon.com/MathScienceTutor/collections

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This section is the hardest section in calculus 1. It's not really that hard after you know how to do it, but at the beginning it feels like rocket science.

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[@badkgaming5122]

@06:35 origin and moving point on graph @11:08 radius of circle is decreasing. How fast is area and circumfuses changing given radius. @16:07 Surface area of a ball decreases at a rate. How fast is diameter changing given radius @19:46 side length of square increases at rate, how fats is area and perimeter changing given side length @23:20 balloon filled with gas at rate, how rate of raidus given radius @27:11 side length of cube increasing at rate, what is rate of surface area and volume given side length @31:11 ladder leaning against house. Slides down wall with rate @46:12 gravel is being dumped at a rate forming a concial pile whos base is 2x height . How fast is height changing given height. @54:48 water is leaking out conical tank at rate, given height and raidus

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This video alone is saving me. I mean, almost every single one of his videos are amazing, but related rates in particular is so difficult to wrap my mind around, and the fact that he included so many examples, is a blessing for real. My teacher only showed us 3 examples, which was really great, but then the homework had examples we hadn't covered yet, and there are so many ways these problems can be written, I'm so glad he went over all these different types. Sincerely, thank you so much.

[@Vyap2702]

my teacher showed 1 example and that's it lmao xed

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True

[@ResidentTitan]

1:19:05 The values of x and y should be switched. x = 120 miles and y = 90 miles since dx/dt = 40 mph (going east) and dy/dt = 30 mph (going north). This makes dz/dt = 50mph. Overall a really great video that helped me understand related rates a lot better. Thank you!

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the hardest part is finding out what you need to find and forming the functions

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Thank you so much! this video is so much better than my proffessors, btw at 1:18:50 I think you might have mixed up x and y. Thanks again

[@patrickong7157]

Yeah he did mix them up.

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I literally came to the comment section to post this same thing. Glad I wasn't going crazy. Changes the answer to 50.

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Best video I have seen on derivatives and related rates by a huge margin. Clear, concise and easy to understand. Thank you. Outstanding!

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[@christianc.7084]

One thing I noticed at 1:21:00 you used the wrong variables. 90 should be multiplied by 30, and 120 by 40. This would change your final answer to be 50 mph, not 48.

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I also noticed this one haha. Its good to double check hehe but still thanks to the video.

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[@harryiguana1761]

at 1:18:03 is there a mistake? Car should be traveling east at 40 not 30 mph

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[@gabriellheureux1294]

correct me if i am wrong but I believe that at 1:18:52 he got the numbers for X and Y mixed up. he has X=90 when It should be 120 and he has Y=120 when it should be 90.

[@JKMath]

You are correct! He did make an error there, x should have been 120 and y should be 90. The rest of his solution is correct, so if you just swap those two values you should get the correct answer! I think the correct answer will be slightly different than his answer. I got 50 when I solved for dz/dt, so I hope this helps! If you ever have any other questions, feel free to stop by my channel and leave me a comment with the question, I would be glad to help you out in any way I can! Have a great day :)

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This video was definitely a life saver, didn't understand this a tap after class and rewatching the class recording, but came here and wow thank you is an understatement!

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[@johnlestergino8645]

In that problem about two cars travelling north and east respectively, x should be 120 and y is equal to 90. In 1:18:30, you used the rate of change of y in getting the value of x which is 30mph instead of 40mph which is the true rate of change of the x. Therefore, the final answer should be 50mph instead of 48mph.

[@arjunshah7138]

Thank you for this comment. I too got 50.

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Just learned this in class and had no fucking idea what my teacher was saying. All my classmates were like “Oh! I get it!” But I was completely dumbfounded. Thank you, really.

[@aldrineuri122]

Same

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You explain this SO well. I was completely confused in my classes, but this video makes so much sense.

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Dude... thank you so much. I thought this impossible but this video made me realize it actually makes complete intuitive sense. For anyone struggling, just follow along with the first few problems and then they get real easy.

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*This man taught me more about related rates in 2 hours then I could've ever imagined, thank you bro*

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[@TheMichaelGer]

Hello, I just need to know at 1:18:56 if you have mistakenly interchanged the variables x and y, where you got x = 90, y = 120 where it should have been x = 120, and y = 90? Thank you

[@JasonVaysberg]

yes. i noticed that too

[@turnertempleton8354]

On the question about the two cars going opposite directions ( 1:18.32 ) I'm pretty sure you messed up by not lining up x and y with dx and dy. This caused you to mess up during the d=Rt portion.

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up

[@iturbo5733]

I was relieved to find I am able to solve these problems in the video successfully after pausing the video when the problem is introduced. I've found that there can be several specific paths to the right answer, all of which are mathematically correct, but can cause some confusion and frustration for those starting out with this section of Calculus I. This is a section that takes a lot of practice to fully grasp and thank goodness for videos like these.

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i just had a related rates quiz today, and your video literally saved my grade!! thank you:)

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[@aaroncortez3826]

1:19:03 after 3 hrs x must be equal to 120mi since the car travels 40 mph east and y must be equal to 90mi as it travels 30 mph north. Am i right ? 🤔

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Aaron Cortez i think you are correct

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I had the same problem mate

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I think you're correct

[@louishennick12]

I thought I did the problem wrong until I saw this comment

[@Slayest_doctor]

@@jonathanmanuales4536 Yeah I noticed that mistake 😅 But he(or she) is a great smart person

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Related rates were challenging for me but you made it seem simple which helped me ace my midterm😊😊

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[@user-mq9ue7tc6r]

119:07 i think you accidentally mixed up the values for x and y, the answer should be 50miles/hour

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Yea I commented this too. Great video though

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I found the same to be true, but the video is still very helpful!

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After watching 10 minutes of this video, I learned more than my teacher had taught me after 6 hours of teaching!

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this is the longest 1hout and a half of my life but it is one of the most important 1 hour and a half to pass Calculus

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Thank you for this lesson you made related rates so much easier for me.

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Correction: 1:22:11 Final answer is 50 miles per hour. He switch/confused the x & y values.

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[@jiminpark4158]

At 20:39 isn’t the dx/dt= +3

[@mandabutangry1146]

came to the comments to make sure I wasn't the only one who noticed this haha

[@陈海鹏-v7b]

I believe u made a mistake for the sample on 1:16:07. You have mis-use the dy/dt as dx/dt, so the distance for yout calculation of X is 30*30=90 miles, but it should be done by used 40*30, and the correct answer for Y should be 120 miles.

[@kriswang2852]

1:18:02 I believe the speed of traveling east is 40m/h instead of 30m/h

[@umamasayed1353]

yes even i believe so , was confused for a sec.

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Thank you for saving my grade. I’m forever grateful

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Wallahi you are a calculus-life saver, I have a big Calc final tomorrow so God bless your soul dude. Thank you!!

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You are a life saver, the way you put helps me formulate what i need to do and then its just algebra with derivatives. This is gonna save me on my AP calc test tomorrow

[@jacott1382]

Thnk you for your video. I just have one question. In your example with 2 cars you choose north as Y and south as X, yet when you calculate the distance you apply dx/dt to north and dy/dt to south.

[@JKMath]

I believe that was an error on his part. x should have been 120 and y should be 90. The rest of his solution is correct, so if you just swap those two values you should get the correct answer! I believe the right answer will be slightly different than his answer. I got 50 when I solved for dz/dt. Hope this helps! If you ever have any other questions, feel free to stop by my channel and leave me a comment with the question, I would be glad to help you out!

[@josh1234567892]

You started losing me after the ladder problem :( The problems get really difficult lmao. Awesome vids though man I really appreciate them!

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I didnt understand this concept even with tutoring and lectures, but I watched 7 minutes of this video and it all made sense. Thanks homie

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I could listen to that soothing voice for hours

[@saeedahamad8928]

Am taking calculus 1 and this part is the hardest one for me I don’t do well with word problems but after watching this video I can’t thank you enough thank you from the bottom of my heart

[@joeyGalileoHotto]

I wish this were around when I was in Calculus 1. I took it a year before this video came out.

[@wallyduboss9464]

I just love this guy's teaching style. Should be a teacher.

[@dominiccirineo4168]

This guy is an absolute hero, A champion of the people, lol. Also, is this Mark Walburg? hahaha

[@BAEX2013G]

You're a rockstar teacher!!! Thank you soooooooo much!