A cylinder is inscribed in a right circular cone of height 8 and radius (at the base) equal to 2.5. What are the dimensions of such a cylinder which has maximum volume?
What is the radius, and height?
To find the dimensions of the cylinder with maximum volume that can be inscribed in the given cone, we need to apply some principles of geometry. Let's break down the problem step by step:
Step 1: Draw a diagram
Draw a right circular cone with a height of 8 units and a radius of 2.5 units.
Step 2: Identify the dimensions of the cylinder
Let the radius of the inscribed cylinder be r, and let the height of the inscribed cylinder be h.
Step 3: Determine the relationship between the cone and the cylinder
The base of the sphere should be tangent to the base of the cone, which means that the radius of the cylinder will be equal to the radius of the cone. Therefore, we can say that r = 2.5.
Step 4: Find the relationship between the heights of the cone and the cylinder
The height of the cylinder should be such that it remains within the cone. Since the height of the cone is 8 units, the height of the cylinder should be less than or equal to 8 units. Let's denote this height as h and as the maximum height the cylinder can have.
Step 5: Use the relationship between the radius and height of the cone to determine h.
By using similar triangles, we can establish a relationship between the radius of the cone, the height of the cone, the radius of the cylinder, and the height of the cylinder. The ratio of the height of the cylinder to the height of the cone will be equal to the ratio of the radius of the cylinder to the radius of the cone.
h / 8 = r / 2.5
Substituting r = 2.5, we can solve for h:
h / 8 = 2.5 / 2.5
h / 8 = 1
h = 8
Step 6: Calculate the volume of the cylinder
Now that we know the radius (r = 2.5) and height (h = 8) of the cylinder, we can calculate its volume using the formula for the volume of a cylinder: V = πr^2h.
V = π(2.5^2)(8)
V = π(6.25)(8)
V = 50π
Therefore, the maximum volume of the inscribed cylinder is 50π cubic units, and its dimensions are a radius of 2.5 units and a height of 8 units.
To find the dimensions of the cylinder with maximum volume inscribed in the given cone, we need to use geometry and calculus.
Let's assume the radius of the cylinder is r and the height of the cylinder is h.
Since the cylinder is inscribed in a right circular cone, the height of the cylinder is equal to the height of the cone. Therefore, h = 8.
To find the radius of the cylinder, we need to consider similar triangles formed by the cone and the radius of the cylinder.
Let's draw a cross-section of the cone and the inscribed cylinder:
```
/|
/ |
r / |
/ |
/ | h
/_____|
Cone
```
We have two similar right triangles: one formed by the radius of the cone (2.5), the height of the cone (8), and the slant height of the cone (s), and another formed by the radius of the cylinder (r), the height of the cylinder (h), and the slant height of the cylinder (s).
Using the property of similar triangles, we can write the following proportion:
r / h = 2.5 / 8
Cross-multiplying, we get:
8r = 2.5h
Substituting h = 8, we have:
8r = 2.5 * 8
Simplifying further:
8r = 20
Dividing by 8, we find:
r = 20 / 8
So the radius of the cylinder is 2.5 units.
Therefore, the dimensions of the cylinder with maximum volume inscribed in the given cone are:
- Radius: 2.5 units
- Height: 8 units