Volume of a cone is 10 cubic cm

[LauraJane]

volume of a cone is 10 cubic cm...

Q: a cone shaped paper drinking cup holds 10 cubic cm of water. We would like to find the height and radius that will require the least amount of paper.

Volume of a cone is: (b x h)/3, or with radius is: ((pi r squared x h))/3.

I think you solve this problem by finding a ratio and getting h in terms of r. Any ideas? Thank you!

 

[amcavoy]

Q: a cone shaped paper drinking cup holds 10 cubic cm of water. We would like to find the height and radius that will require the least amount of paper.

Volume of a cone is: (b x h)/3, or with radius is: ((pi r squared x h))/3.

I think you solve this problem by finding a ratio and getting h in terms of r. Any ideas? Thank you!

You know:

$$\text{V}=\frac{\pi\,r^{2}\,h}{3}=10$$

If I remember correctly, the surface area (what you want to minimize) is:

$$\text{S}=\pi\,r\,\sqrt{r^{2}+h^{2}}$$

Now minimize this taking into consideration the restraint on volume.

 

[quicknote]

I would like to clarify that the given information does not provide enough data to accurately determine the height and radius of the cone. The volume of a cone is dependent on both the height and radius, and in order to find the minimum amount of paper required, we would need to know the exact shape and dimensions of the cone.

However, in general, to minimize the surface area of a cone, we can use the calculus method of finding the critical points. This involves finding the derivative of the surface area equation with respect to the variable we are trying to minimize (in this case, either the height or radius) and setting it equal to 0. Then, we can solve for the variable and plug it back into the equation to find the minimum surface area.

Another approach could be to experiment with different values of height and radius and calculate the surface area for each combination until we find the one with the minimum amount of paper. However, this method may not be as accurate and efficient as using calculus.

In conclusion, without more information, it is not possible to accurately determine the height and radius of the cone that would require the least amount of paper. Further experimentation or calculations using calculus may be necessary to find the optimal solution.