Need help with volumes of spheres with holes.

In summary, the conversation discusses the volume of napkin rings created by drilling holes through two wooden balls of different diameters. The question is posed whether one ring has more wood in it than the other. It is suggested to use the Pythagorean theorem to find the relationship between the radius of the sphere, the radius of the hole, and the height of the napkin ring. The conversation includes a visual representation of this relationship and discusses how it can be used to solve the problem.
  • #1
timm3r
10
0

Homework Statement


tim1.jpg

if you guys have trouble reading that, it says,

Suppose you make napkin rings by drilling holes with different diameters through two wooden balls (which also have different diameters). You discover that both napkin rings have the same height h, as shown in the figure.
a) Guess which ring has more wood in it.
b) Check your guess: use cylindrical shells to compute the volume of a napkin ring created by drilling a hole with radius r through the center of a sphere of radius R and express the answers in terms of h.


Homework Equations


I know how to find the volume of this using discs, which the equation would be
integral[-R, R] sqrt[R^2 - X^2] - integral[-r, r] sqrt[R^2 - x^2]


The Attempt at a Solution


i have no idea how to answer this in terms of h, can someone please help me put this in terms of h.
 
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  • #2
There is a pretty simple relation between h, the radius of the sphere, and the radius of the hole that follows from pythagoras' theorem.
 
  • #3
so if we use the pythagoras theorem, 2r would be the relation between the radius of the hole, what would be the relation of the radius of the sphere?
 
  • #4
timm3r,

Draw a triangle with one vertex at the center of the sphere, the 2nd at the top wall of the cylinder, and the 3rd straight down from the second so that it forms a 90 degree angle with the first two. What is the length of the 3 sides of the triangle? That should give you your relationship between R, r, and h.
 
  • #5
im still stuck on this problem, can someone make it more dumby proof for me?
 
  • #6
ok i still don't get it, this is what i did
tim2.jpg

i still don't see how there is a relation between the radius of the circles.
 
  • #7
It's hard to make it any clearer. One side of that triangle you drew is h/2, one is the radius of the sphere, and one is the radius of the hole. It's a right triangle, so you can use pythagoras' law.
 
  • #8
ok did you see my picture? so if we go back to the c^2=a^2+b^2, a would be h/2, b would be the radius of the the hole, and c would be the radius of the sphere right?
 
  • #9
timm3r said:
ok did you see my picture? so if we go back to the c^2=a^2+b^2, a would be h/2, b would be the radius of the the hole, and c would be the radius of the sphere right?
How about writing that in the terms you are given: If r is the radius of the hole and R is the radius of the sphere, then [itex]r^2+ (h/2)^2= R^2[/itex]. Since h, and so h/2, is the same in both spheres, we have R^2- r^2 the same in both spheres.
 

1. How do you calculate the volume of a sphere with a hole?

The volume of a sphere with a hole can be calculated by subtracting the volume of the hole from the volume of the entire sphere. The formula for the volume of a sphere with radius r and a hole with radius rh is: V = (4/3)πr3 - (4/3)πrh3.

2. What is the purpose of calculating the volume of a sphere with a hole?

Calculating the volume of a sphere with a hole is useful in many engineering and scientific applications. It can help determine the capacity of containers with cylindrical cavities, such as pipes and water tanks, or the amount of material needed to fill a spherical mold with a hollow center.

3. How do you find the radius of the hole in a sphere with a known volume?

The radius of the hole in a sphere can be found by rearranging the formula for the volume of a sphere with a hole. The formula for the radius of the hole is: rh = r((3V) / (4π))1/3.

4. Can the volume of a sphere with a hole be negative?

No, the volume of a sphere with a hole cannot be negative. The volume of any object is a measure of the amount of space it occupies, and space cannot have a negative value.

5. How do you convert the volume of a sphere with a hole from cubic units to other units of measurement?

To convert the volume of a sphere with a hole from cubic units (such as cubic meters or cubic inches) to other units of measurement, you can use conversion factors. For example, to convert from cubic meters to cubic centimeters, multiply the volume in cubic meters by 1,000,000 (1 cubic meter = 1,000,000 cubic centimeters).

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