Question

Find the largest volume of a cylinder that fits into a cone that has base radius rr and height h.?

Answer 1

The largest volume of a cylinder that fits into a cone can be found by setting the volume of the cylinder equal to the volume of the fluid it displaces.

Step-by-step explanation:

The largest volume of a cylinder that fits into a cone can be found by setting the volume of the cylinder equal to the volume of the fluid it displaces.

If the base radius of the cone is and the height is , then the volume of the cylinder is given by the formula: V = πr²h

To find the largest volume, we can take the area of the cone (A = πr²) and subtract the volume of a smaller cylinder in the cone (V = πr²(h₂ - h₁)). This will give us the maximum volume of the cylinder that fits into the cone.

Answer 2

The largest volume of a cylinder that can fit into a cone with base radius r and height h is
`\( (4)/(27)\pi hr^2 \).`

To find the largest volume of a cylinder that fits into a cone with a base radius r and height h we need to follow these steps:

Step 1: Understand the Problem

We have a right circular cone with radius r and height h We need to find the largest possible cylinder that can fit inside this cone. The cylinder will also be right circular and its dimensions will be constrained by the cone's dimensions.

Step 2: Set Up the Equations

Let's denote:

- The radius of the cylinder as
`\( r_c \).`

- The height of the cylinder as
`\( h_c \).`

Since the cylinder is inside the cone, the top of the cylinder will form a similar triangle with the cone. The ratio of the sides of these similar triangles gives us:

`\[ (r_c)/(h - h_c) = (r)/(h) \]`

From this, we can express
`\( r_c \) in terms of \( h_c \):`

`\[ r_c = (r)/(h) * (h - h_c) \]`

Step 3: Express the Cylinder's Volume

The volume V of the cylinder is given by:

`\[ V = \pi r_c^2 h_c \]`

Substitute
`\( r_c \)` from the earlier equation:

`\[ V = \pi \left((r)/(h) * (h - h_c)\right)^2 h_c \]`

Step 4: Maximize the Volume

To find the maximum volume, we need to differentiate this volume equation with respect to
`\( h_c \)` and set the derivative equal to zero.

`\[ (dV)/(dh_c) = 0 \]`

Step 5: Solve for
`\( h_c \)`

By solving this equation, we can find the value of
`\( h_c \)`that maximizes the volume. This involves some calculus.

Step 6: Find
`\( r_c \)`and the Maximum Volume

Once we have
`\( h_c \), we can find \( r_c \)`using the similar triangles ratio. Then, we substitute
`\( r_c \) and \( h_c \)`back into the volume equation to find the maximum volume.

Now, let's perform the necessary calculations.

The critical points for the height of the cylinder
`\( h_c \) inside the cone are \( (h)/(3) \)` and h . The second solution, h , is not valid as it would imply a cylinder of the same height as the cone, which is not possible due to the tapering of the cone. Therefore, the valid solution is
`\( h_c = (h)/(3) \).`

Step 7: Calculate
`\( r_c \)`and the Maximum Volume

Now, let's calculate the radius of the cylinder
`\( r_c \) using \( h_c = (h)/(3) \)` and then find the maximum volume of the cylinder.

`\[ r_c = (r)/(h) * \left(h - (h)/(3)\right) = (2)/(3)r \]`

And the maximum volume V is:

`\[ V = \pi \left( (2)/(3)r \right)^2 * (h)/(3) \]`

Let's calculate the maximum volume.

The radius
`\( r_c \)` of the cylinder that maximizes its volume inside the cone is
`\( (2)/(3)r \)`, and the maximum volume V of this cylinder is
`\( (4)/(27)\pi hr^2 \).`