Related Rates | Gravel Cone Pile Height

In this video, we use related rates to determine the rate of change of the heigh of a gravel cone pile.

#calculus #relatedrates

[samg]

I haven’t done this sort of math in 20 years and listening to this is like trying to remember a fever dream that I can almost remember but can’t put my finger on it

[bend.manevitz]

I did it differently, and I can't tell if they're the same at some deeper level, or if your way is more readily made general...

V = (π•h³)/4 = (π/4)•h³
V also = 30t (t is in minutes)

Set those equal to each other and you have h as a function of t:

h(t) = ³√(360t/π). Evaluate at h=10 and t=(25π/9) = ~8.73.

Take the derivative of h(t) and you get a hot mess that smarter people than i could probably do, but I went to Wolfram Alpha. Got something cool but complicated with fractional exponents, but appeoximates to:

h`(t)=~ 1.61906/(t^(2/3))

Plugged in t=(25π/9) from above and got 0.382

[_dzdtsunami]

Rate of volume = 30 cubic ft per min
Height = 10ft

Derivative of volume = rate of volume (+)
Derivative of height = rate of height (+)

Simple reversing operations were used to find rate of height increase.

[googleyoutubeaccount]

I haven’t got a clue what you’re talking about but thanks for trying 😂 Video is well made mate, good work

[bogdangiusca]

It grows by 0.37 feet/minute.

Explanation
We start with a cone of hight of 10 ft
And a radius of 5ft

Volume of a come is pi*r^2*h÷3= 261.80 ft^3

Next we add the new volume of 30 ft^3
For a new total of 291.80 ft^3

Now we do the volum requation in revers to try and find out the new hight.

We trun r into h/2 (because the diameter is equal to the hight)

Also instead of h i will use X to better understand this is a new hight.

Pi*(X/2)^2*X÷3=291.80

X^3=291.80×3×4÷pi
X^3= 1114.60 ft^3
X= 10.37 ft
Now to find the grow rate we just substract the original hight of 10.
We will call the growth rate G and its duration is per minute.

G=X-h=10.37-10=0.37 ft/min

[OSAS]

Did you rewrite the “r” for radius in terms of height “h”because you were trying to eliminate the “r” from the conic formula? Because to me this looks a lot like implicit differentiation. And it looks like you chained out the h’ from the h^2 square using the chain rule in order to isolate the dh/dt or (h’) on one side of the differential equation. Am I correct?

[shanebassen]

But the equation does not take into account that the pile is growing. The growth will require more gravel for every foot it goes up.