Problem Analysis

We need to find the altitude $h$ of an inscribed cone within a larger cone such that the inscribed cone's volume is maximized. Here's the information provided:

• The larger cone has:

  • Volume $V = 9 \, \text{m}^3$,
  • Altitude $H = 4 \, \text{m}$.

• The inscribed cone shares the same axis as the larger cone, and its vertex is at the base of the larger cone.

Step-by-Step Solution

1. Volume formula for a cone

The volume of a cone is given by: $V = \frac{1}{3} \pi r^2 h,$ where:

• $r$ is the radius of the cone's base,
• $h$ is the altitude of the cone.

2. Relating dimensions of the inscribed cone to the larger cone

• Let the inscribed cone's altitude be $h$, where $0 \leq h \leq 4$.
• By similar triangles, the radius $r$ of the inscribed cone's base is proportional to its altitude: $\frac{r}{h} = \frac{R}{H},$ where $R$ and $H$ are the radius and altitude of the larger cone. Substituting $R = 3 \, \text{m}$ (calculated from $V = \frac{1}{3} \pi R^2 H = 9$) and $H = 4$: $\frac{r}{h} = \frac{3}{4} \implies r = \frac{3}{4} h.$

3. Volume of the inscribed cone

Substitute $r = \frac{3}{4} h$ into the cone's volume formula: $V_{\text{inscribed}} = \frac{1}{3} \pi \left( \frac{3}{4} h \right)^2 h = \frac{1}{3} \pi \cdot \frac{9}{16} h^3 = \frac{3 \pi}{16} h^3.$

4. Maximize the volume

To maximize $V_{\text{inscribed}} = \frac{3 \pi}{16} h^3$, we differentiate with respect to $h$: $\frac{dV_{\text{inscribed}}}{dh} = \frac{9 \pi}{16} h^2.$

Set $\frac{dV_{\text{inscribed}}}{dh} = 0$ to find critical points: $\frac{9 \pi}{16} h^2 = 0 \implies h = 0 \quad \text{or} \quad h = 4.$

Since $h = 0$ gives zero volume, and $h = 4$ corresponds to the maximum possible inscribed volume, the answer is:

$\boxed{4 \, \text{m}}$

Let me know if you'd like additional clarification or have further questions!

Follow-Up Questions

1. How does the ratio of radii between the cones relate to maximizing the volume?
2. Why is the altitude $h$ of the inscribed cone directly proportional to the outer cone's altitude?
3. Can the optimization result change for different outer cone volumes?
4. How can we generalize this problem for cones with arbitrary dimensions?
5. How does calculus (specifically differentiation) ensure the maximum volume is found?

Tip

Always confirm the relationship between similar triangles when working with inscribed figures; it often simplifies the problem!