The task is to calculate the largest possible volume of a cone given that the slant height (образующая) is $12.9$ cm.

Formula for Volume of a Cone:

$V = \frac{1}{3} \pi r^2 h$

Where:

• $r$ is the radius of the cone's base.
• $h$ is the height of the cone.
• The slant height $l = 12.9$ cm relates to $r$ and $h$ via the Pythagorean theorem: $l^2 = r^2 + h^2.$

Strategy for Maximum Volume:

To maximize $V$, we express $h$ in terms of $r$ and substitute into the volume formula: $h = \sqrt{l^2 - r^2}.$

The volume becomes: $V = \frac{1}{3} \pi r^2 \sqrt{l^2 - r^2}.$

Constraint:

Under the square root, the expression $l^2 - r^2$ must remain non-negative, i.e., $r \leq l$.

Step-by-Step Solution:

1. Substitute $l = 12.9$ into the equation: $h = \sqrt{12.9^2 - r^2}.$

2. Substitute this into the volume formula: $V = \frac{1}{3} \pi r^2 \sqrt{12.9^2 - r^2}.$

3. Maximize $V$. This requires calculus or observation:

   • The function $r^2 \sqrt{12.9^2 - r^2}$ reaches a maximum when $r^2 = \frac{l^2}{3} = \frac{12.9^2}{3}$.
   • Calculate $r^2 = \frac{166.41}{3} = 55.47$.
   • The corresponding $h = \sqrt{12.9^2 - r^2} = \sqrt{166.41 - 55.47} = \sqrt{110.94}$.

4. Substitute $r^2$ into $V$:

   • $V = \frac{1}{3} \pi \cdot 55.47 \cdot \sqrt{110.94}$.

Let me calculate this for you. math

Given values

l = 12.9 # slant height in cm

Calculate r^2 and h for maximum volume

r_squared = l2 / 3 # r^2 = l^2 / 3 h = math.sqrt(l2 - r_squared) # height h

Calculate volume V

V = (1/3) * math.pi * r_squared * h

r_squared, h, VThe calculations yield the following results:

1. $r^2 = 55.47$ (radius squared),
2. $h = 10.53 \, \text{cm}$ (height of the cone),
3. $V = 611.83 \, \text{cm}^3$ (maximum volume of the cone).

Thus, the largest possible volume of the cone is approximately $611.83 \, \text{cm}^3$.

Would you like further details on the calculations or have any other questions? Here are some related questions for exploration:

1. How does the slant height affect the volume of a cone?
2. How can we derive the formula for the volume of a cone geometrically?
3. Why does maximizing $r^2\sqrt{l^2 - r^2}$ yield the maximum volume?
4. Can the volume be maximized using a different method, such as numerical analysis?
5. How would the result change if the slant height were a different value?

Tip: Always double-check your calculations when maximizing or minimizing functions using derivatives or approximations.