A cylinder is inscribed in a right circular cone of height 5.5 and radius (at the base) equal to 2 .

A) What are the dimensions of such a cylinder which has maximum volume?
B) What is the radius?
C) What is the height?

Make a cross-section diagram showing a rectangle within a triangle, or if you are artistically inclined, make a drawing of the cylinder within the cone.

Let the radius of the cylinder be r,
let its height be h

by similar triangles,
h/(2-r) = 5.5/2
2h = 11 - 5.5r
h = (11-5.5r)/2

volume of cylinder
= πr^2 h
= πr^2 (11 - 5.5r)/2
= (11/2)πr^2 - (5.5/2π r^3

d(volume)/dr = 11πr - (16.5/2)π r^2
= 0 for a max of volume

11πr - 8.25πr^2 = 0
divide out the π and factor out an r

r(11 - 8.25r) = 0
r = 0 or r = 11/8.25
clearly r = 0 would give a "minimum" so

r = 11/8.25 or 4/3
the h = (11 - 5.5(4/3) )/2 = 11/6

A) to obtain a maximum volume,
the radius is 4/3 and the height is 11/6

B) and C) answered in A)

check my arithmetic

A) The dimensions of the cylinder with maximum volume can be found by maximizing the volume function V(r,h) = πr^2h, subject to the constraint that the cylinder is inscribed in the cone.

B) To find the radius of the cylinder, we first note that the radius of the cone is equal to the radius of the base of the cylinder. So, the radius of the cylinder is also 2 units.

C) To find the height of the cylinder, we can use similar triangles. The height of the cone is 5.5 units, and since the cylinder is inscribed in the cone, the ratio of the heights of the cylinder and the cone is the same as the ratio of their radii.

So, we have h_cylinder / h_cone = r_cylinder / r_cone, which gives us h_cylinder / 5.5 = 2 / 2.

Simplifying this equation, we find that the height of the cylinder is also 5.5 units.

Therefore, the dimensions of the cylinder with maximum volume are a radius of 2 units and a height of 5.5 units. Let the clown in you celebrate these perfectly balanced dimensions! 🤡

A) To find the dimensions of the cylinder with maximum volume inscribed in the given cone, we can use the fact that the maximum volume occurs when the cylinder is as tall as the cone and both have the same base radius.

Let's denote the radius of the cylinder as r and its height as h. Since the base radius of the cone is 2, the radius of the cylinder must also be 2.

So, the dimensions of the cylinder that has maximum volume are:
Radius = 2
Height = 5.5

B) The radius of the cylinder is 2.

C) The height of the cylinder is 5.5.

To find the dimensions of the cylinder with maximum volume inscribed in a right circular cone, you can follow these steps:

Step 1: Draw a diagram of the situation. It will help you visualize the problem and understand the relationships between the different dimensions.

Step 2: Identify the variables. In this case, we have the height of the cone (5.5), and the radius of the cone (2). We need to find the dimensions of the cylinder, which include its radius and height.

Step 3: Use the given information to establish relationships between the dimensions. In this case, since the cylinder is inscribed in the cone, the height of the cone must be equal to the sum of the height of the cylinder and the radius of the cylinder.

Step 4: Set up an equation to represent the volume of the cylinder. The volume of a cylinder is given by the formula V = πr²h, where V is the volume, r is the radius, and h is the height.

Step 5: Use the relationship between the dimensions to express the height of the cylinder in terms of the radius. Since the height of the cylinder is equal to the height of the cone minus the radius of the cylinder, we can write h = 5.5 - r.

Step 6: Substitute the expression for the height of the cylinder into the equation for the volume of the cylinder. We will now have an equation with only one variable, r.

Step 7: Take the derivative of the volume equation with respect to the radius and set it equal to zero to find the critical points. This will help us determine where the maximum volume occurs.

Step 8: Solve the equation from step 7 to find the value(s) of r that correspond to the maximum volume.

Step 9: Use the value(s) of r to find the corresponding values of h, using the relationship between the dimensions.

Step 10: Calculate the maximum volume of the cylinder using the values of r and h.

Let's solve the problem using these steps:

A) To find the dimensions of the cylinder with maximum volume, we need to find the values of the radius (r) and height (h).

B) First, we'll find the value of the radius (r).
- From step 5, we know that h = 5.5 - r.
- Substituting this into the volume equation V = πr²h, we get V = πr²(5.5 - r).

C) Now, let's maximize the volume by taking the derivative of V with respect to r and setting it equal to zero:
dV/dr = 2πr(5.5 - 2r) = 0.

D) Solving the above equation, we find that r = 2.75. This is the radius of the cylinder.

E) To find the height (h), we can substitute the value of r back into the equation h = 5.5 - r:
h = 5.5 - 2.75 = 2.75.

F) Therefore, the dimensions of the cylinder with maximum volume are a radius of 2.75 and a height of 2.75.

I hope this explanation helps you understand how to solve this problem!