Related Rates #2 Using Cones

🪨 Master Related Rates with the Gravel Pile Problem! 🪨

In this informative calculus video, we explore a related rates problem involving a pile of gravel. As gravel is dumped into the pile, we’ll determine how quickly the height of the pile is changing. Understanding this concept is essential for applying related rates in real-world scenarios!

What You’ll Learn:

Understanding Related Rates: Grasp the fundamentals of related rates problems in calculus.

Step-by-Step Approach: Follow along as I guide you through the process:
Creating a Diagram: Visualize the gravel pile to clarify the relationship between variables.
Labeling Rates: Identify and label all relevant rates of change.
Finding the Equation: Establish the equation that relates the volume of the gravel pile to its height.
Taking the Derivative: Differentiate the equation with respect to time (d/dt) to relate the rates.
Substituting Values: Plug in specific information to solve for the rate at which the height is changing.

Why Watch This Video?

Ideal for Students: Perfect for high school and college students studying calculus and related rates.
Clear Explanations: Easy-to-follow instructions that simplify complex calculus concepts.
Enhance Your Problem-Solving Skills: Build confidence in tackling related rates problems in calculus.

📈 Don’t Forget to:

LIKE this video if you find it helpful!
SHARE with classmates or friends who want to master related rates!
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[patrickjmt]

Hi all! Wanna help a Youtube education OG? Please post comments, questions and anything else on your mind in the comment section! so, don’t forget to LIKE, THUMBS UP, and SUBSCRIBE! I’d appreciate it greatly as it helps me :)

[doorduude]

If anyone else was confused why the radius is (h/2)^2 when he re-writes the equation, its because you're given that the height = diameter. So half of the diameter = the radius. So when you re-write the equation instead of having r^2 you have (h/2)^2. This part is easy but took me a long time to visualize. Hopefully this helps one of you.

[hxctrae]

I've watched tons and tons of your videos... and I just now realized...

You're left-handed!
I hope me being left-handed as well gives me an edge on my AP Calculus BC exam TOMORROW ! Thanks for all of your help, Patrick! You've made this an easier year in school for me. I'll spread the word of your legendary skills.

[vanessarahimi]

Just to clear things up. You only use the product rule when you are multiplying 2 VARIABLES. For example, you would use the product rule for (4x)(3x). However, for his example, V=pi/12 * h ^3 is NOT multiplying two variables. In math, Pi is considered a constant, so pi/12 is also a CONSTANT, not a variable. h^3 is a variable, and it is being multiplied by the constant pi/12, so we cannot use the power rule. Also, yes the derivative of a constant is 0, but only if that constant is being added or subtracted. For the example in the video, he is MULTIPLYING a *constant* and a *variable*, the constant is not being added or subtracted! In situations like this, you leave the constant alone, take the derivative of the variable, and then just multiply it by the constant. For example, the derivative of 7x^3 is (7) (3x^2) which when multiplied is equal to 21x^2. Hope this helps

[FrostByte_AC]

My only problem is, I practice these (like I should) and the bitch'll slap a number "e" on the exam. BAM, down the drain. That area of a cone is very tricky too.

[purpk]

Patrick, you never cease to amaze me.
My friend and I are were stuck on this very exact problem in our homework except that the rate was 30 instead of 20ft/min. I see the light!!! xoxoxox from Los Angeles

[jessieskates]

Thank you so much for your videos. I just finished a LATE night study sesh with them, and they helped ever so much! I know I'll be watching more of them as my math career goes on! Thank you again!!

[Jono.]

I find it hilarious that this problem is for all intensive purposes completely identical to the stuff im working on 8 years later. Thank you for the video, helpful as always!

[noorzaid]

thank you for all of you videos, I will never finish school without them.

[patrickjmt]

@deis7 no, height is not measured in cubic feet. i am not 6 cubic feet tall.

[DAsahutube]

This video was great! One thing that I think is an easy throw off it the height divided by 2. Since the height and diameter are equal than the h/2 and radius are equal too. Just to clear things up.

[dborked]

Great video. You're a lifesaver, I just had a homework problem exactly like this with different numbers. So much easier to understand than my professor. Thanks a bunch!

[Wesb]

WOW! I didn't even think of using h to represent d, that makes finding r^2 so much simpler! Thank you! I never would have figured out my own problem without thinking of that!

[marioman]

Thank you! It helps that your handwriting is really neat. Every time my classmates point to their work and explain, I don't see what's going on. Really, thanks. It's MUCH easier than I thought.

[patrickjmt]

ya, you got to use the correct formula! : )
glad the rest helped too!

[anomnomnomous]

I missed the day in class when we were taught this topic. These helped sooo much. Thank you, Patrick!

[angcientrock]

in the beginning, you are given that the height and diameter are equal. By substituting h for r there are less variables to deal with. Because the volume formula uses radius, PJMT had to write it as h/2 (d=2r, r=d/2)

[Harcher]

I have the highest grade in my calculus class due to your videos. Thanks

[rampant]

polishing up my stuff for final, this helped :)

[sammyhtb]

thanks for helping me graduate high school and helping me in college right now!