What is the optimal angle for a conical cup with a maximum volume?

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In summary, the question is asking for the angle @ that will result in the maximum volume of a conical cup made by joining the edges OA and OB of a sector of a circle with a radius of 8cm. The volume formula is given as V=pi*r^2*h/3 and the height, h, needs to be expressed in terms of @. This can be done by using similar triangles and solving for h as h=1/(1-R/r). However, the value of r is not given and a picture would help in understanding the problem better.
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cateater2000
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A conical cup is to be made by joining the edges OA, OB of the sector of a circle of radius 8cm. What angle @ gives the cup of maximum volume?


I'm having trouble solving this question. This is what I have done so far

we know V=pi*r^2*h/3

r is given, so we need to put h in terms of @.
now by similar triangles we can take another slice, and compare
r/h=R/(h-1)... (h-1) is just the height of the triangle below the new slice
solving for h I get h=1/(1-R/r) now I'm unsure of what to do now, and how to get @ into the equation.


Anyways please let me know if I'm doing this question completely wrong or what not. Thanks for all your help
 
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  • #2
r is not given! You're given the radius of the sector of the circle but that is not the same as the radius of the cone.
 
  • #3
a picture would help
 

Related to What is the optimal angle for a conical cup with a maximum volume?

1. How do I calculate the volume of a cone?

The formula for calculating the volume of a cone is V = (π * r^2 * h) / 3, where r is the radius of the base and h is the height of the cone.

2. What is the maximum volume of a cone?

The maximum volume of a cone occurs when the height is equal to the radius of the base, resulting in a volume of (π * r^3) / 3.

3. How do I maximize the volume of a cone?

To maximize the volume of a cone, you need to find the height that will result in the maximum volume. This can be done by taking the derivative of the volume formula with respect to height, setting it equal to 0, and solving for h.

4. Can the volume of a cone be negative?

No, the volume of a cone cannot be negative. It is a measure of the amount of space inside the cone and therefore must be a positive value.

5. What are some real-life applications of maximizing the volume of a cone?

Maximizing the volume of a cone can be useful in various engineering and construction projects, such as designing efficient water tanks or silos. It can also be applied in optimizing the storage capacity of ice cream cones or other conical-shaped containers.

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