Final answer:
To find the ratio of the altitude to the base radius of the cone of least possible volume, we need to consider the relationship between the cone and the circumscribed sphere. By differentiating the volume formula and finding the minimum volume, we can determine the ratio as 2:1.
Step-by-step explanation:
To find the ratio of the altitude to the base radius of the cone of least possible volume, we need to consider the relationship between the cone and the circumscribed sphere. First, let's define the variables:
- r = radius of the sphere
- h = altitude of the cone
- R = base radius of the cone
- V = volume of the cone
Since the cone is circumscribed about the sphere, the height of the cone is equal to the diameter of the sphere. Therefore, h = 2r. The volume of the cone is given by the formula V = (1/3)πR²h. Substituting h = 2r, we get V = (4/3)πR²r.
Now, to find the cone of least possible volume, we need to minimize V with respect to R and r. To do this, we can use calculus. We differentiate V with respect to R, set it equal to zero to find the critical values, and then substitute those values back into V to find the minimum volume. The final expression for the volume of the cone of least possible volume is V = (4/3)πr³.
Therefore, the ratio of the altitude to the base radius of the cone of least possible volume is 2r: r, which simplifies to 2:1.