A cylinder is inscribed in a right circular cone of height 6.5 and radius (at the base) equal to 5.5. What are the dimensions of such a cylinder which has maximum volume?
To find the dimensions of the cylinder that has the maximum volume inscribed in a right circular cone, we need to consider the geometry of the problem.
Let's define the dimensions of the cylinder:
- Let the height of the cylinder be 'h'
- Let the radius of the base of the cylinder be 'r'
We need to maximize the volume of the cylinder. The volume of a cylinder is given by the formula:
V = π * r^2 * h
We have the following conditions:
1. The cylinder is inscribed in a right circular cone, meaning that the base of the cylinder is the same as the base of the cone.
2. The height of the cone is 6.5 and the radius of the cone at the base is 5.5.
Let's assume the vertex of the cone lies directly above the center of the base of the cylinder. In this case, the height of the cylinder will be equal to the height of the cone.
Therefore, we have:
h = 6.5
Now, let's consider the radius of the cylinder. Since the cylinder is inscribed in the cone, the radius of the cone at the base is equal to the diagonal of the rectangle that can be formed by the radius of the cylinder and half the height of the cylinder.
Using the Pythagorean theorem, we can find the relationship between the radius of the cone, the radius of the cylinder, and half the height of the cylinder.
We have:
(radius of cone)^2 = (radius of cylinder)^2 + (0.5 * height of cylinder)^2
(5.5)^2 = r^2 + (0.5 * h)^2
Substituting the known values, we have:
(5.5)^2 = r^2 + (0.5 * 6.5)^2
r^2 = (5.5)^2 - (0.5 * 6.5)^2
r^2 = 30.25 - 10.5625
r^2 = 19.6875
Taking the square root of both sides, we find:
r ≈ 4.44
Therefore, the radius of the cylinder that maximizes its volume is approximately 4.44. And since the height of the cylinder is equal to the height of the cone, the height of the cylinder is 6.5.
Hence, the dimensions of the cylinder that has the maximum volume inscribed in the given right circular cone are approximately:
- Radius = 4.44
- Height = 6.5
in the cylinder, upside down.
diameter=5.5(6.5-h)/6.5
volume=PI*h*D^2/4
V=PI/4*h*(5.5/6.5)^2*(6.5-h)
V=PI/4*(5.5/6.5)^2 (6.5h-h^2)
dV/dh=0=6.5-2h
h=3.25
d=5.5/6.5(3.25)=5.5/2=2.75