Consider the right circular cone shown. if the radius of the circular base remains the same and the height varies, what is the minimum surface area and minimum volume the cone can have?
Copy and paste does not work. No cone is shown.
However, since volume of a cone = (1/3) * pi * radius^2 * height, as long as radius remains the same, the minimal height will = minimum volume. The same with surface area.
22/7×98/100
To find the minimum surface area and minimum volume of a cone when the radius of the circular base remains the same and the height varies, we need to understand the relationship between the surface area, volume, radius, and height of a cone.
The formula for the surface area of a cone is:
S = πr^2 + πrℓ
where S is the surface area, r is the radius of the circular base, and ℓ is the slant height.
The formula for the volume of a cone is:
V = (1/3)πr^2h
where V is the volume, r is the radius of the circular base, and h is the height.
Now, let's consider the minimum surface area and minimum volume of the cone:
1. Minimum Surface Area:
To minimize the surface area, we need to minimize both the curved surface area (πrℓ) and the area of the base (πr^2).
Since the radius of the circular base remains the same, we cannot change its value. However, we can minimize the slant height ℓ by making it as small as possible. The minimum slant height occurs when the cone becomes a flat triangle with the height approaching zero. In this case, the curved surface area (πrℓ) will also be zero, and the only area left is the area of the base, which is πr^2. Therefore, the minimum surface area is πr^2.
2. Minimum Volume:
To minimize the volume, we need to minimize the height while keeping the radius of the circular base the same. As the height approaches zero, the volume will also approach zero. Therefore, the minimum volume is zero.
In summary, when the radius of the circular base remains the same and the height varies:
- The minimum surface area is πr^2.
- The minimum volume is zero.