Find the dimensions of a cylinder of maximum volume that can be contained within, and share the axis of symmetry of, a square regular pyramid with a height of 15 cm and a base with side 6 cm

To find the dimensions of the largest cylinder that can fit inside a square pyramid and shares the axis of symmetry with the pyramid, we need to:

1. **Analyze the Pyramid**:
   - Height \( h_p = 15 \) cm
   - Base side length \( a_p = 6 \) cm
   - The axis of symmetry is vertical, through the center of the base and the apex of the pyramid.

2. **Cylinder's Dimensions**:
   - Radius of the cylinder's base \( r \)
   - Height of the cylinder \( h_c \)

### Step 1: Equation of the Pyramid’s Slant Height
We can derive the equation of the pyramid's side profile to help determine the maximum radius of the cylinder at different heights.

- The half-width of the pyramid at the base is \( \frac{a_p}{2} = 3 \) cm.
- The slant height of the pyramid decreases linearly as you move up from the base to the apex.

The relationship between the width of the pyramid \( w(y) \) at a height \( y \) (measured from the base upwards) and the height is given by:
\[
w(y) = \frac{6}{15}(15 - y) = 6 - \frac{2}{5}y
\]
Thus, the radius \( r(y) \) of the cylinder at height \( y \) must be half of \( w(y) \):
\[
r(y) = \frac{w(y)}{2} = 3 - \frac{1}{5}y
\]

### Step 2: Volume of the Cylinder
The volume \( V \) of the cylinder can be expressed as:
\[
V = \pi r^2 h_c
\]
Substituting the radius function \( r(y) = 3 - \frac{1}{5}y \), we get:
\[
V = \pi (3 - \frac{1}{5}y)^2 y
\]

### Step 3: Maximizing the Volume
To find the value of \( y \) that maximizes the volume, we take the derivative of the volume with respect to \( y \) and set it to zero.

1. Expand the volume equation:
   \[
   V(y) = \pi \left(9 - \frac{6}{5}y + \frac{1}{25}y^2\right) y = \pi \left(9y - \frac{6}{5}y^2 + \frac{1}{25}y^3\right)
   \]

2. Differentiate \( V(y) \) with respect to \( y \):
   \[
   \frac{dV}{dy} = \pi \left(9 - \frac{12}{5}y + \frac{3}{25}y^2\right)
   \]

3. Set the derivative equal to zero to find the critical points:
   \[
   9 - \frac{12}{5}y + \frac{3}{25}y^2 = 0
   \]

Multiply the entire equation by 25 to eliminate fractions:
\[
225 - 60y + 3y^2 = 0
\]
This is a quadratic equation:
\[
3y^2 - 60y + 225 = 0
\]
Simplify by dividing by 3:
\[
y^2 - 20y + 75 = 0
\]
Solve using the quadratic formula:
\[
y = \frac{-(-20) \pm \sqrt{(-20)^2 - 4(1)(75)}}{2(1)} = \frac{20 \pm \sqrt{400 - 300}}{2} = \frac{20 \pm \sqrt{100}}{2} = \frac{20 \pm 10}{2}
\]
Thus, \( y = 15 \) cm or \( y = 5 \) cm.

### Step 4: Determine Maximum Volume
The cylinder must fit inside the pyramid, so the cylinder's height cannot exceed the height of the pyramid. Hence, the height \( y = 5 \) cm corresponds to the cylinder.

### Step 5: Dimensions of the Cylinder
At \( y = 5 \) cm:
- Radius: \( r(5) = 3 - \frac{1}{5}(5) = 2 \) cm
- Height: \( h_c = 5 \) cm

### Final Answer:
The cylinder of maximum volume that fits within the pyramid has:
- Radius = 2 cm
- Height = 5 cm

Would you like more details on any part of the solution?

---

Here are five related questions:
1. How would the volume change if the base of the pyramid had a different side length?
2. What is the surface area of the maximum volume cylinder within the pyramid?
3. Can we fit a cone instead of a cylinder inside the pyramid? How would the dimensions differ?
4. How does the formula for the volume of a cylinder compare to that of a cone?
5. What if the pyramid had a rectangular base? How would the approach change?

**Tip**: For optimization problems, always check that your solution satisfies the geometric constraints.

Find the dimensions of a cylinder of maximum volume that can be contained within, and share the axis of symmetry of, a square regular pyramid with a height of 15 cm and a base with side 6 cm.

Explore how to determine the maximum volume of a cylinder that can fit inside a square pyramid with a height of 15 cm and a base side of 6 cm. This problem involves geometry and optimization, using calculus to find the critical points. Suitable for advanced high school 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