Find the Minimum Volume of cone, answer does not make sense

In summary: The top of the sphere would be tangent to the side of the cone. In summary, the dimensions of the right circular cone of minimum volume that can be circumscribed about a sphere of radius 8 inches are: height = 32/3 inches, base radius = 16/3 inches. The minimum volume of the cone is approximately 123.46 cubic inches.
  • #1
Brown Arrow
101
0

Homework Statement



Find the Dimensions of the right circular cone of minimum volume which can be circumscribed about a sphere of radius 8 inches.

Homework Equations



N/A

The Attempt at a Solution


So this is my try, i did the question to find the minimum volume of the cone,
2e2ibm9.jpg

For Larger Size: http://i54.tinypic.com/29yqyiw.jpg
As you can see the minimum value is when h=0 , thus volume is equal to 0, it makes sense the minimum volume would be 0, but i think i did the question wrong, can you tell me where I've gone wrong.

Please help
 
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  • #2
Your picture shows a cone that is inscribed inside a sphere, not circumscribed outside as the problem asks.
 
  • #3
so is it suppose to be like this then i try to minimize the volume of the cone,
mimqgo.jpg

sorry i didn't get what it meant mean circumscribed , i though it would be inside
 
  • #4
Brown Arrow said:

Homework Statement



Find the Dimensions of the right circular cone of minimum volume which can be circumscribed about a sphere of radius 8 inches.

Homework Equations



N/A

The Attempt at a Solution


So this is my try, i did the question to find the minimum volume of the cone,
2e2ibm9.jpg

For Larger Size: http://i54.tinypic.com/29yqyiw.jpg
As you can see the minimum value is when h=0 , thus volume is equal to 0, it makes sense the minimum volume would be 0, but i think i did the question wrong, can you tell me where I've gone wrong.

Please help

Your calculation (given your drawing) is completely correct.
It's only your graphic interpretation with a parabola that is incorrect.
The volume is not given by a parabola but by a third power.
So the maximum volume is at h=32/3.
Btw, the volume becomes zero at h = 16, which is the solution for V=0.
 
  • #5
Brown Arrow said:
so is it suppose to be like this then i try to minimize the volume of the cone,
mimqgo.jpg

sorry i didn't get what it meant mean circumscribed , i though it would be inside

Yes. But your picture needs a little work. The bottom of the sphere would touch the center of the base of the cone.
 

1. What is the formula for finding the minimum volume of a cone?

The formula for finding the minimum volume of a cone is V = (1/3)πr²h, where V is the volume, r is the radius, and h is the height.

2. How do you use this formula to find the minimum volume of a cone?

To find the minimum volume of a cone, you would need to know the radius and height of the cone. Plug these values into the formula V = (1/3)πr²h and solve for V.

3. Can the minimum volume of a cone be negative?

No, the minimum volume of a cone cannot be negative. The volume of a cone is always a positive value.

4. How does the minimum volume of a cone relate to its shape?

The minimum volume of a cone is directly related to its shape. A cone with a larger radius and height will have a larger minimum volume, while a cone with a smaller radius and height will have a smaller minimum volume.

5. Can you explain what "minimum volume" means in the context of a cone?

In the context of a cone, "minimum volume" refers to the smallest possible volume that a cone can have. This would occur when the cone has the smallest possible radius and height while still maintaining its shape.

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