Using differentiation to prove

[casey123]

Homework Statement

'Use differentiation to prove that as the radius of the base of a cone increases, the height of the cone decreases.'

Homework Equations

The height has been previously calculated as:

h = (507 - 2pi r^3) / pi r^2

The Attempt at a Solution

I have worked out that:

h' = (-1014 - 2pi r^3) / pi r^3

I'm just not sure how to do the proving section though. I can do it without differentiating by replacing values for r in the h equation and also by graphing the h equation, but that doesn't answer the question as differentiation is not used. Any help would be greatly appreciated!

 

[chaoseverlasting]

Since the cone has constant volume, you can write it out as
$$V=\frac{\pi r^2h}{3}$$. Now, you can write out r and h in terms of the angle of the cone (that the curve makes with the base), differentiate the expression for the volume and show that its a monotonic function and use the relation b/w r and h to prove your result.

 

[HallsofIvy]

Homework Statement

'Use differentiation to prove that as the radius of the base of a cone increases, the height of the cone decreases.'

Well, that's NOT true is it? Perhaps you forgot to add "a cone with constant volume".

Homework Equations

The height has been previously calculated as:

h = (507 - 2pi r^3) / pi r^2

Where did you get that formula? Where is the "507" from? That equation is the same as $\pi r^2 h+ 2\pi r^3= 507$. The first term of that is 3 times the volume of a cone (or the volume of a cylinder) but I don't recognize the second term (except, perhaps, as (3/4) the volume of a sphere.)

The Attempt at a Solution

I have worked out that:

h' = (-1014 - 2pi r^3) / pi r^3

I'm just not sure how to do the proving section though. I can do it without differentiating by replacing values for r in the h equation and also by graphing the h equation, but that doesn't answer the question as differentiation is not used. Any help would be greatly appreciated!

Saying that "as the radius of the base of a cone (with constant volume?) increases, the height of the cone decreases" means that the derivative of the height, with respect to the radius, must be negative.

 

[casey123]

Yes, it is assuming the volume of the cone is constant. (Thanks for pointing that out!) The question is based around an application concerning ice-cream cones. The first part was to do with maximising the volume of ice-cream that could fit into a cone. The height expression was derived from those calculations.

I think i may have been a bit off track though, after reading your replies. Am I correct in saying that this question (of proving height decreases as radius increases) should concern only the volume of the cone, not the volume i have used (which includes the half-sphere of ice-cream on top)?

Do I then need to manipulate the volume expression so that it is in terms of h and r only, then find the derivative? If this derivative expression is negative is that proof that h decreases as r increases?

Sorry if I've confused anyone, and thanks again helping me. I really need it!