Calculating the Diameter of a Drilled Hole in a Revolved Solid

In summary, the problem involves finding the diameter of a hole in a solid generated by revolving the region bounded by y = x^2/2 and y = 2 around the y-axis. After using cylindrical shells to integrate and finding the overall volume to be 4pi/3, it is determined that one-fourth of the volume is removed after drilling the hole. The volume of the hole is then found to be pi, and it is determined to be a right cylindrical shape. After some calculations and corrections, the correct diameter of the hole is found to be 21/2.
  • #1
icesalmon
270
13

Homework Statement


A solid is generated by revolving the region bounded by y = x2/2 and y = 2 around the y-axis. A hole, centered along the axis of revolution, is drilled through this solid so that one-fourth of the volume is removed. Find the diameter of the hole.

The Attempt at a Solution


I'm going with cylindrical shells this time around. I'm integrating from x = 0 to x = 2. I think my height is 2-x and my radii are all going to be generated by the function x2/2. After integration, I get the overall volume is to be 4pi/3. If one quarter of that is taken out after drilling this hole, I have the volume of this figure to be pi. I believe it's a right cylindrical shaped whole, so the volume of a cylinder is pir2h. I need the radius of one of these cross sections so I need some way to relate the volume. Or maybe I don't, this is where I need assistance. Hopefully I'm not over-thinking this. Thanks.
 
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  • #2
I believe I've made a mistake on the radius / height. Radius = x and Height = x2/2, after integration I get vt = 4pi so the volume of the cylinder is 3pi.
 
  • #3
icesalmon said:
I believe I've made a mistake on the radius / height. Radius = x and Height = x2/2, after integration I get vt = 4pi so the volume of the cylinder is 3pi.
Yes, except that the removed core is not exactly a cylinder. What do you get for the diameter of the hole?
 
  • #4
haruspex said:
Yes, except that the removed core is not exactly a cylinder. What do you get for the diameter of the hole?
I don't know how to find it, especially since I don't know what the figure of the hole is.

Edit: the diameter seems like it would be 2(y/2)1/2 but that's sort of a shot in the dark.
Edit: that's wrong, nvm.
 
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  • #5
icesalmon said:
I believe I've made a mistake on the radius / height. Radius = x and Height = x2/2, after integration I get vt = 4pi so the volume of the cylinder is 3pi.
The height of each shell should be 2 - x2/2 .

Do the same integral but have x go from 0 to a. set that volume to 1/4 the volume without the hole & solve for a .
 
  • #6
I'm getting two answers..2 and 2(21/2)
and neither are correct.

Should have included this, sorry. My integrand here has been changed to (x)(2-x2/2). When my bounds are [0,a] I get 2pi(a2 - a4/8). When my bounds are [0,2] I get the volume as 4pi. If I set the first part equal to 1/4 of the total volume I get 2pi(a2 -a4/8) = pi and I solve it from there to get that a = 21/2
 
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  • #7
icesalmon said:
2pi(a2 -a4/8) = pi
Right
a = 21/2
Wrong.
 
  • #8
thanks, got it now.
 

Related to Calculating the Diameter of a Drilled Hole in a Revolved Solid

1. What tools do I need to measure the diameter of a hole?

To find the diameter of a hole, you will need a measuring tape or ruler, a caliper, or a drill bit gauge. These tools will allow you to accurately measure the diameter of the hole.

2. How do I measure the diameter of a hole?

To measure the diameter of a hole, place the measuring tool across the center of the hole and take the measurement from one edge to the other. If using a caliper or drill bit gauge, ensure that the measuring arms are properly aligned with the center of the hole.

3. What is the unit of measurement for the diameter of a hole?

The diameter of a hole is typically measured in either inches or millimeters. The unit of measurement will depend on the size and precision of the hole being measured.

4. Can I use a ruler to measure the diameter of a hole?

While a ruler can be used to measure the diameter of a hole, it may not be the most accurate method. A ruler may not have fine enough increments to measure small or precise holes. It is recommended to use a caliper or drill bit gauge for more accurate measurements.

5. How do I know if my measurement of the hole diameter is accurate?

To ensure the accuracy of your measurement, it is recommended to take multiple measurements using different tools. If all measurements are consistent, then you can be confident in your measurement. If there are discrepancies, try taking the measurement again or seek the assistance of a more precise measuring tool.

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