A right circular cylinder of radius r and height h is inscribed in a right circular cone of radius R and height H, as shown on the right figure. Find the value of r (in terms of R and H) that maximizes the total surface area of the cylinder (including top and bottom). Comment on the influence of the relationship between R and 2H to the max/min of total surface area.
To find the value of r that maximizes the total surface area of the cylinder, we first need to understand the relationship between the dimensions of the cone and the cylinder.
Let's consider the right circular cylinder inscribed in the right circular cone. The cylinder touches the cone at two points - its base touches the base of the cone, and its top touches the curved surface (slant height) of the cone.
Now, let's analyze the dimensions of the figures involved:
1. Radius of the cylinder: r
2. Height of the cylinder: h
3. Radius of the cone: R
4. Height of the cone: H
In this configuration, the following relationships hold:
1. The height of the cylinder (h) is equal to the height of the cone (H).
2. The radius of the base of the cylinder (r) and the radius of the base of the cone (R) are parallel and have a constant difference.
To proceed, we need to express the total surface area of the cylinder in terms of r, R, and H.
The total surface area of the cylinder is given by the sum of the areas of the two circular bases and the curved surface area:
Total surface area of the cylinder = 2 * (area of base) + (curved surface area)
The area of the base of the cylinder is given by π * r^2.
The curved surface area of the cylinder is represented by the rectangular shape when the cylinder is unrolled into a flat surface. The dimensions of this rectangle are the circumference of the cylinder's base (2πr) and the height of the cylinder (h). Therefore, the curved surface area is given by 2πrh.
Now, we can express the total surface area of the cylinder in terms of r, R, and H:
Total surface area = 2 * π * r^2 + 2 * π * r * h
Substituting h with H (height of the cone) in the equation, we get:
Total surface area = 2 * π * r^2 + 2 * π * r * H
To find the value of r that maximizes the total surface area, we can take the derivative of the total surface area equation with respect to r, set it equal to zero, and solve for r.
d(Total surface area)/dr = 4πr + 2πH = 0
Simplifying the equation, we get:
4πr + 2πH = 0
Dividing both sides by 2π, we obtain:
2r + H = 0
Therefore, the value of r that maximizes the total surface area is given by:
r = -H/2
Now, we need to comment on the influence of the relationship between R and 2H on the maximum/minimum of the total surface area of the cylinder.
When considering the relationship between R and 2H, if R becomes smaller compared to 2H, the base of the cone gets narrower, resulting in a smaller space for the cylinder to be inscribed. As a result, the maximum surface area of the cylinder decreases.
On the other hand, if R becomes larger compared to 2H, the base of the cone becomes wider, providing more space for the cylinder to be inscribed. This leads to an increase in the maximum surface area of the cylinder.
In summary, when 2H becomes larger compared to R, the maximum surface area of the cylinder also increases. Conversely, when R becomes larger compared to 2H, the maximum surface area of the cylinder decreases.