Rate of increase of a radius and height of a cone

AI Thread Summary
The discussion focuses on the relationship between the height and radius of a conical pile of coal, defined by the angle of repose of 38%. The formula h = r * tan(a) is established to express height in terms of radius. A volumetric rate of change of 0.3 m³/min is provided, prompting the need to determine the rate of increase of the radius when it measures 1.7 m. Participants suggest deriving the cone's volume formula and differentiating it to find the relationship between the rates of change. Understanding these relationships is crucial for solving the problem effectively.
brandy
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Homework Statement


when powder or granular solids are piled up. the powder forms a conical pile. the edge of the pile reaches a certain maximum angle with the horizontal, called the angle or repose.

A) a pile of coal is found to have an angle of repose of 38%
what is the relationship between height of the pile and its radius.

i got that bit: h=r*Tan a
h= height, r=radius a= Angle of repose


B)if a conveyor belt pours the coal onto the pile at a rate of 0.3m^3/min, at what rate is the radius increasing when the radius is 1.7m

i think i might have to derive the equaiton but i really don't know. i think i need the formula for the area of cone but i don't know. if i did i wouldn't be posting here!
 
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hi brandy

you are given a volumetric rate of change (dV/dt) and want to know the rate of change of the radius (dr/dt).

Can you write the volume of a cone in terms of its radius? You will probably have to use your results form a) to write the height in terms of the radius as well.

Then you can look at differentiating to relate the rates
 
A cone has a third of the volume of a cylinder with the same height and base. Can you find the volume of the cone with that?
 
I tried to combine those 2 formulas but it didn't work. I tried using another case where there are 2 red balls and 2 blue balls only so when combining the formula I got ##\frac{(4-1)!}{2!2!}=\frac{3}{2}## which does not make sense. Is there any formula to calculate cyclic permutation of identical objects or I have to do it by listing all the possibilities? Thanks
Since ##px^9+q## is the factor, then ##x^9=\frac{-q}{p}## will be one of the roots. Let ##f(x)=27x^{18}+bx^9+70##, then: $$27\left(\frac{-q}{p}\right)^2+b\left(\frac{-q}{p}\right)+70=0$$ $$b=27 \frac{q}{p}+70 \frac{p}{q}$$ $$b=\frac{27q^2+70p^2}{pq}$$ From this expression, it looks like there is no greatest value of ##b## because increasing the value of ##p## and ##q## will also increase the value of ##b##. How to find the greatest value of ##b##? Thanks

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