How to find which z-value to cut at to get the appropriate volume of a sphere

In summary, the conversation is about finding the z-coordinates to make a horizontal cut in a unit sphere to get a specific volume cut off. The speaker has a basic understanding of calculus and is seeking help with finding the solution.
  • #1
Pixel08
3
0
Hi PF!

I've been trying to find out how one could find which z-coordinates to 'cut' at to get a specific volume of the sphere cut off.

i.e. I have a unit sphere, therefore the total volume is (4/3)*pi*r^3, where r = 1. Now I want to get 1/4 of that volume cut off. So, (1/4)*(4/3)*pi*r^3.

But the problem is, how do I find out where I made that single cut? (The cut has to be horizontal).
 
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  • #2
What is your mathematics background?
 
  • #3
DivisionByZro said:
What is your mathematics background?

Hi DivisionByZro! I'm a first year student, just started post-secondary. Not much of a mathematics background - took a differential calculus course last term.
 
  • #4
What do you know about integrals?
 
  • #5


I would suggest using the formula for the volume of a spherical cap, which is V = (1/3)*pi*h^2*(3r-h), where h is the height of the cap and r is the radius of the sphere. In this case, we know the total volume of the sphere and we want to find the height of the cap that would give us 1/4 of that volume. So, we can rearrange the formula to solve for h:

h = r - √(3V/(4pi*r))

Plugging in the values for a unit sphere and 1/4 of its volume, we get:

h = 1 - √(3*(1/4)*(4/3)*pi*1) = 1 - √(3/3) = 1 - 1 = 0

This means that the height of the cap is 0, which makes sense since we want a horizontal cut. Therefore, the z-coordinate of the cut would be 0, and the volume of the spherical cap would be 1/4 of the total volume of the sphere.

In summary, to find the appropriate z-coordinate for a specific volume of a spherical cap, we can use the formula V = (1/3)*pi*h^2*(3r-h) and rearrange it to solve for h, which represents the height of the cap. This method can be applied to any desired volume, not just 1/4 of the total volume.
 

Related to How to find which z-value to cut at to get the appropriate volume of a sphere

1. What is a z-value and how does it relate to finding the appropriate volume of a sphere?

A z-value is a statistical measure that represents the number of standard deviations a data point is above or below the mean. In the context of finding the appropriate volume of a sphere, the z-value is used to determine the appropriate radius to cut the sphere at in order to achieve the desired volume.

2. How do I calculate the z-value for a specific volume of a sphere?

The z-value can be calculated using the formula z = (V - V0) / σ, where V is the desired volume, V0 is the mean volume of a sphere, and σ is the standard deviation of the sphere's volumes. This formula can be found in most statistical textbooks or online resources.

3. Can the z-value be negative when cutting a sphere to achieve a specific volume?

Yes, the z-value can be negative if the desired volume is smaller than the mean volume of the sphere. This would indicate that the radius needs to be cut by a certain amount to decrease the volume to the desired amount.

4. Are there any other factors besides the z-value that need to be considered when cutting a sphere to achieve a specific volume?

Yes, the z-value is just one factor to consider. Other factors that may need to be taken into account include the shape and density of the sphere, as well as any potential measurement errors.

5. Can the z-value be used to cut a sphere to achieve a non-standard volume, such as a fraction or multiple of the mean volume?

Yes, the z-value can be used to calculate the appropriate radius for any desired volume, whether it is a fraction or multiple of the mean volume. However, it is important to note that this may result in a non-ideal shape for the sphere, and other factors may need to be considered in order to achieve the desired volume accurately.

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