Find the maximum volume of right circular cylinder that can be inscribed in a cone of altitude 12 cm and base radius 4 cm, if the axes of the cylinder and con coincide.

Try to make a sketch of a cylinder inside a cone

Draw in the altitude, let the height be h
let the radius of the cylinder be r
Look at a cross section of the diagram.
the altitude from the top of the cylinder to the vertex of the cone is 12-h
and by similar triangles
(12-h)/r = 12/4 = 3/1
3r = 12-h
h = 12-3r

V(cylinder) = πr^2 h
= πr^2 (12-3r)
= 12πr^2 - 3πr^3
dV/dr = 24πr - 9πr^2 = 0 for a max of V
3πr(8 - 3r) = 0
r = 0 , clearly yielding a minimum Volume
or
r = 8/3

max V = ....

(you do the button pushing)

Well, well, well, let's see here. We have a cone and we want to inscribe a cylinder in it. That's like putting a clown hat on a cone, but instead of a hat, it's a cylinder. I must say, that would be quite the fashion statement!

Now, to maximize the volume of the cylinder, it needs to perfectly fit inside the cone. Imagine two clowns trying to fit inside a tiny clown car, it's all about maximizing the space!

The base of the cylinder will be the same as the base of the cone, so we know its radius is 4 cm. The height of the cone is given as 12 cm, which means the height of the cylinder will also be 12 cm.

Now, let's calculate the volume of the cylinder using the formula V = πr²h. Plug in the values we have: V = π(4 cm)²(12 cm). That gives us a volume of 192π cm³.

So, the maximum volume of the cylinder that can be inscribed in the cone is 192π cm³. That's a lot of clown magic packed into a cone!

To find the maximum volume of a right circular cylinder inscribed in a cone, we need to first determine the height and radius of the cylinder.

1. Consider the cone:
- Altitude of the cone (h) = 12 cm
- Base radius of the cone (r) = 4 cm

2. Since the cylinder is inscribed in the cone, its height and radius will be equal to the altitude and radius of the cone at the point of contact.

3. The relationship between the height (h1) and radius (r1) of the cylinder and the height (h) and radius (r) of the cone is given by similar triangles:

h1 / r1 = h / r

4. Substituting the given values:
h1 / r1 = 12 / 4
h1 / r1 = 3

5. We know that the volume (V) of a right circular cylinder is given by V = π * r^2 * h.

6. Substitute the relationship from step 4:
V = π * r1^2 * h1
V = π * r1^2 * 3r1
V = 3π * r1^3

7. To maximize the volume, we need to find the maximum value of r1. Since r1 cannot exceed the radius of the cone (r), the maximum value of r1 is 4 cm.

8. Substitute r1 = 4 into the volume equation:
V = 3π * 4^3
V = 3π * 64
V = 192π cm^3

Therefore, the maximum volume of the inscribed cylinder is 192π cubic centimeters.

To find the maximum volume of a right circular cylinder that can be inscribed in a cone, we need to consider the relationship between the dimensions of the cylinder and the cone.

First, let's define the variables:
- Vc: Volume of the cylinder
- Vco: Volume of the cone
- rc: Radius of the cylinder
- hc: Height of the cylinder
- rco: Radius of the cone
- hco: Height of the cone

In this case, the axis of the cylinder and the cone coincide, which means that the height of the cylinder (hc) is equal to the height of the cone (hco).

The volume (V) of a right circular cylinder is given by the formula:
Vc = π * rc^2 * hc

The volume (V) of a cone is given by the formula:
Vco = (1/3) * π * rco^2 * hco

We need to find the maximum value for Vc, so we need to relate the radius and height of the cylinder to the radius and height of the cone.

Since the cylinder is inscribed in the cone, the base of the cylinder must touch the base of the cone, which means that the radius of the cylinder (rc) must be equal to the radius of the cone (rco).

Now, we have:
rc = rco

We also know that hc = hco (since the axes of the cylinder and the cone coincide).

Therefore, we can rewrite the formulas as:
Vc = π * rc^2 * hc
Vco = (1/3) * π * rc^2 * hc

Since rc = rco and hc = hco, we can simplify the formulas to:
Vc = π * rco^2 * hco
Vco = (1/3) * π * rco^2 * hco

To find the maximum volume of the cylinder, we need to maximize Vc.

To maximize the volume, we take the derivative of Vc with respect to rco and set it equal to zero:
dVc/drco = 2π * rco * hco = 0

Solving this equation, we find that rco = 0, which is not possible in this case since both the radius of the cone and the cylinder must be positive.

Therefore, there is no maximum volume for the cylinder that can be inscribed in this cone.

In other words, the maximum volume of a right circular cylinder that can be inscribed in a cone of altitude 12 cm and base radius 4 cm, when the axes of the cylinder and cone coincide, is "undefined" or "does not exist".