Find the largest volume of the right circular cylinder that can be inscribed in a right circular cone with base radius 6 cm and height 10 cm.


To solve this problem, we will use calculus to maximize the volume of the right circular cylinder inscribed in a right circular cone.

### Step-by-Step Solution

1. **Understand the Geometry**:

   - We have a right circular cone with a base radius \( R = 6 \) cm and height \( H = 10 \) cm.
   - A right circular cylinder is inscribed within this cone. Let the radius of the cylinder be \( r \) and the height be \( h \).

2. **Relationship Between Cylinder and Cone**:

   To find a relationship between \( r \) and \( h \), consider a vertical cross-section of the cone and the inscribed cylinder. The cross-section forms two similar triangles:
   
   - The large triangle represents the cross-section of the cone, with base \( 2R = 12 \) cm and height \( H = 10 \) cm.
   - The smaller triangle represents the cross-section of the cylinder, with base \( 2r \) and height \( h \).

   By similarity of triangles:

   \[
   \frac{r}{h} = \frac{R - r}{H - h}
   \]

   Substituting \( R = 6 \) and \( H = 10 \):

   \[
   \frac{r}{h} = \frac{6 - r}{10 - h}
   \]

   Cross-multiplying gives:

   \[
   r(10 - h) = h(6 - r)
   \]

   Simplifying, we get:

   \[
   10r - rh = 6h - hr
   \]

   Rearranging terms:

   \[
   10r = 6h
   \]

   Solving for \( r \) in terms of \( h \):

   \[
   r = \frac{3h}{5}
   \]

3. **Volume of the Cylinder**:

   The volume \( V \) of a cylinder is given by:

   \[
   V = \pi r^2 h
   \]

   Substitute \( r = \frac{3h}{5} \):

   \[
   V(h) = \pi \left(\frac{3h}{5}\right)^2 h = \pi \frac{9h^3}{25}
   \]

   Simplify:

   \[
   V(h) = \frac{9\pi h^3}{25}
   \]

4. **Maximize the Volume**:

   To maximize \( V(h) \), we take the derivative with respect to \( h \) and set it to zero:

   \[
   V'(h) = \frac{d}{dh} \left( \frac{9\pi h^3}{25} \right) = \frac{27\pi h^2}{25}
   \]

   Set \( V'(h) = 0 \):

   \[
   \frac{27\pi h^2}{25} = 0
   \]

   Solving for \( h \), we find:

   \[
   h = 0 \quad \text{or} \quad h = \frac{10}{3}
   \]

   Since \( h = 0 \) gives zero volume, we choose \( h = \frac{10}{3} \) cm.

5. **Find the Corresponding Radius**:

   Now, find the radius \( r \) corresponding to \( h = \frac{10}{3} \):

   \[
   r = \frac{3h}{5} = \frac{3 \times \frac{10}{3}}{5} = 2 \, \text{cm}
   \]

6. **Calculate the Maximum Volume**:

   Substitute \( h = \frac{10}{3} \) and \( r = 2 \) into the volume formula:

   \[
   V = \pi r^2 h = \pi (2)^2 \left( \frac{10}{3} \right) = \pi \cdot 4 \cdot \frac{10}{3}
   \]

   Simplify:

   \[
   V = \frac{40\pi}{3} \, \text{cm}^3
   \]

### Conclusion

The largest volume of the right circular cylinder that can be inscribed in the right circular cone is:

\[
\boxed{\frac{40\pi}{3} \, \text{cm}^3 \approx 41.89 \, \text{cm}^3}
\]

Would you like more details or have any questions?

### Further Questions to Explore:
1. How does the maximum volume change if the cone's height is increased while keeping the base radius constant?
2. What is the largest surface area of the cylinder that can be inscribed in the cone?
3. How would you find the maximum volume of a cylinder inscribed in a cone with an elliptical base?
4. Can you derive a general formula for the maximum volume of an inscribed cylinder in any cone?
5. How does the optimal radius \( r \) relate to the cone's dimensions?

### Tip
To maximize or minimize geometric quantities, always look for relationships between dimensions using similarity or other geometric properties.

Find the largest volume of the right circular cylinder that can be inscribed in a right circular cone with base radius 6 cm and height 10 cm.

Discover how to find the largest possible volume of a cylinder inscribed in a cone with a base radius of 6 cm and a height of 10 cm. This problem involves calculus, geometry, and optimization techniques, making it ideal for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation