Find largest possible volume (Extreme Value)

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A right triangle the hypotenuse of which is c revolves about a small side. A right circular cone is formed. Find the largest possible volume of the cone.

Vcone: pi*r^2*(h/3)

I don't get why they mentioned the hypotenuse, since it is not even in the volume formula...
Since it is a 'right' triangle, shouldn't the maximum height of the cone be 1?
But I still don't know which extreme value I should search for...
(The solution should be: 1/(9*sqrt3) * c^3 * pi )

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fresh_42
Mentor
A right triangle the hypotenuse of which is c revolves about a small side. A right circular cone is formed. Find the largest possible volume of the cone.

Vcone: pi*r^2*(h/3)

I don't get why they mentioned the hypotenuse, since it is not even in the volume formula...
Since it is a 'right' triangle, shouldn't the maximum height of the cone be 1?
But I still don't know which extreme value I should search for...
(The solution should be: 1/(9*sqrt3) * c^3 * pi )

You mentioned the solution which obviously has a ##c## in it. So probably the hypotenuse will play a role.
How do you derive that ##h=1##?
As I see it from your solution you are supposed to maximize ##V_{cone}## for a given length of the hypotenuse ##c##.
Clearly ##c=0## is a minimum.
How do you usually calculate maxima?

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We only learned to make the derivative equal to zero in order to find it.

pbuk
Gold Member
I think there is a mistake in the answer which you will find if you work through it.

If you know that the sloping side of the cone is the hypotenuse of a right angled triangle length c, can you find a relationship between c, r and h?

isn't it
c^2= r^2+h^2 ?
r^2 = c^2 - h^2

Then I could substitute it:

V = pi * (c^2-h^2) * h/3
then
V = pi * ( (c^2h/3) - (h^3/3) )
V = pi * h/3 * (c^2 - h^2)

derivative:

pi * h/3 * d/dc (c^2 - h^2)
= 2/3 * pi * h * c

But this isn't the solution... What did I do wrong?

HallsofIvy
Homework Helper
The fact that they say "the hypotenuse of which is c" tells you that the hypotenuse is a constant. You certainly should NOT be differentiating with respect to c!

pbuk
Gold Member
isn't it
c^2= r^2+h^2 ?
r^2 = c^2 - h^2

Then I could substitute it:

V = pi * (c^2-h^2) * h/3
then
V = pi * ( (c^2h/3) - (h^3/3) )
You're doing fine up to here, let me write it more clearly as $$V = \frac{\pi c^2}3 h - \frac 13 h^3$$So taking on board HallsOfIvy's comment, what's next?

Ray Vickson
Homework Helper
Dearly Missed
You're doing fine up to here, let me write it more clearly as $$V = \frac{\pi c^2}3 h - \frac 13 h^3$$So taking on board HallsOfIvy's comment, what's next?

No. The equation
$$V = \frac{\pi c^2}3 h - \frac 13 h^3$$
is not correct. However, the equation
$$V = \frac{\pi}{3} \left( c^2 h - h^3 \right)$$
is correct.

pbuk
$$V = \frac{\pi c^2}3 h - \frac 13 h^3$$
$$V = \frac{\pi}{3} \left( c^2 h - h^3 \right)$$