A cylinder is inscribed in a right circular cone of height 6.5 and radius (at the base) equal to 6. What are the dimensions of such a cylinder which has maximum volume?
To find the dimensions of the cylinder with maximum volume inscribed in a cone, we need to set up a mathematical equation. Let's denote the radius of the cylinder as r and the height as h.
1. Since the cylinder is inscribed in the cone, its height will be equal to the height of the cone, which is 6.5.
2. The radius of the cylinder will be equal to the radius of its base, which in this case is equal to 6.
3. To maximize the volume of the cylinder, we need to find an equation that represents the volume in terms of a single variable, either r or h. Since the radius is given, let's express the volume in terms of r.
4. The volume of a cylinder is given by the formula V = π * r^2 * h. Substituting the given values, we have V = π * 6^2 * 6.5.
5. To maximize the volume, we can take the derivative of the volume equation with respect to r and set it equal to zero. This will allow us to find the critical points where the volume is at its maximum.
6. Differentiating the volume equation with respect to r, we get dV/dr = 2πrh. Equating this to zero, we have 2πrh = 0.
7. Since the height h is positive, the equation becomes 2πr = 0. Solving for r, we find that the radius of the cylinder is equal to zero.
8. However, a cylinder with zero radius does not have any volume. Hence, this critical point does not represent the maximum volume.
9. We also need to check the endpoints of the possible range for r. In this case, the radius of the cone's base is 6, so the maximum possible radius for the cylinder is 6.
10. By evaluating the volume at the endpoints r = 0 and r = 6, as well as the critical point r = 0, we can determine which one gives the maximum volume.
11. Evaluating the volume equation at r = 0 gives V = π * 0^2 * 6.5 = 0.
12. Evaluating the volume equation at r = 6 gives V = π * 6^2 * 6.5 = 468π.
13. Comparing the volumes obtained at the endpoints, we can see that V = 468π is greater than V = 0. Therefore, the maximum volume is achieved when the radius of the cylinder is 6.
So, the dimensions of the cylinder with maximum volume inscribed in the given cone are:
- Radius (r) = 6
- Height (h) = 6.5
there are lots of solutions to this online. A good one is at
http://answers.yahoo.com/question/index?qid=20080427160136AACsLX6
where they show that the largest such cylinder has volume 4/9 the volume of the cone. That is, when the cylinder has 1/3 the height of the cone.