A cylinder is inscribed in a right circular cone of height 8 and radius (at the base) equal to 8. What are the dimensions of such a cylinder which has maximum volume?

Let r be the radius of the inscribed cylinder. The top edge of the cylinder (where the height is h) must touch the cone, where h and r are related by the straight-line equation

h = 8 - r.
The volume of the cone is
V (r) = pi*r^2 h = pi*r^2*(8-r)
= pi*[8 r^2 - r^3]

When the cylinder's volume is a maximum, dV/dr = 0, so
16r = 3r^2
r = 16/3, which is 2/3 the height of the come.

To find the dimensions of the cylinder with maximum volume inscribed in a right circular cone, we can use the concept of similar triangles. Let's denote the radius of the cylinder as r and its height as h.

Step 1: Draw the diagram
Let's draw a right circular cone with height 8 and radius 8 at the base. Now, draw a cylinder with its base coinciding with the base of the cone.

Step 2: Determine the similar triangles
Notice that the cross-section of the cone and the cylinder are similar triangles. This means that the corresponding sides of these triangles are proportional.

Step 3: Write the similarity ratio
Let's write the similarity ratio between the cone and the cylinder. Since the ratio of corresponding sides in similar triangles is the same, we can write:

(r / 8) = (h / 8)

Step 4: Solve for r in terms of h
Rearrange the equation to solve for r:

r = (h / 8) * 8
r = h

Step 5: Write the volume formula for the cylinder
The volume of a cylinder is given by the formula:

V = π * r^2 * h

Substitute the value of r from step 4:

V = π * (h^2) * h
V = πh^3

Step 6: Maximize the volume
To find the dimensions of the cylinder with maximum volume, we need to maximize the volume formula. Since h is the only variable in the formula, we can find the maximum value of the volume by maximizing h.

Step 7: Determine the maximum value of h
Since the height of the cone is 8, the maximum value of h should not exceed 8. Therefore, the maximum value of h is 8.

Step 8: Calculate the dimensions of the cylinder with maximum volume
Substitute the maximum value of h into the formula for r:

r = 8

Therefore, the dimensions of the cylinder with maximum volume are:
- Radius (r): 8
- Height (h): 8

To find the dimensions of the cylinder with maximum volume, we need to first understand the geometry of the problem.

Let's consider the following diagram:

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r

In the diagram, we have a right circular cone with height h and radius (at the base) also denoted as r. The cylinder that we are interested in is inscribed inside this cone.

The volume of a cylinder is given by V = πr^2h, where r is the radius of the cylinder's base and h is its height. To find the dimensions of the cylinder with maximum volume, we need to find the values of r and h that will maximize this equation.

Since the cylinder is inscribed in the cone, it means that the height of the cylinder is equal to the height of the cone. Therefore, h = 8.

Now, let's focus on the radius of the cylinder's base, which we will denote as R. To find R, we need to use similar triangles.

By looking at the diagram, we can see that the ratio of the radius of the cone's base (r) to the height of the cone (h) is constant. This means that r/h is equal to a constant value, let's call it k.

So, we have r/h = k.

Substituting h = 8, we get r/8 = k.

Now, let's find the value of k. We know that when r = 8 (the radius of the cone's base), h = 8 (the height of the cone). Therefore, we can write:

8/8 = k.

Simplifying, we find that k = 1.

So, now we know that r/h = 1.

Since we have r = R (the radius of the cylinder's base) and h = 8 (the height of the cylinder), we can write R/8 = 1.

Simplifying, we find that R = 8.

Therefore, the dimensions of the cylinder with maximum volume are:
- Radius of the cylinder's base (R) = 8
- Height of the cylinder (h) = 8

So, the cylinder with maximum volume has a radius of 8 and a height of 8.