I obtained this one from a book. Above is the illustration. There is a plug carved out of cork so that it can fit in each of the triangular, circular, and square holes. The bottom surface is a circle, and the top edge is a straight line, equal to the the diameter of the circle and parallel to the plane of the circle. The curved surface is generated by lines reaching from the circumference of the circle to the upper line. The lines lie in parallel planes, which in turn are perpendicular to the circle. Of course, the illustration shows only a subset of the generating lines. Hopefully by now you have the picture.
This week’s Quiz Question is: what is the volume of the plug in terms of the diameter of the circle? I did not look at the answer in the book, but I am sure I have figured this one out. The book says it’s not necessary to use calculus to compute the volume. My method uses only methods of solid geometry, which course I took in high school.
Post your answer as a comment below.
Update and solution
I am first going to post my solution. Then I’m going to the book and have a look at their solution. Some explanation:
The statement of the problem in the book says that you don’t need calculus to work this problem. If you take a look at it your first temptation might be to integrate over the enclosed shape and derive the volume. That would be using integral calculus. You would evaluate an integral such as this one:
That’s the something I will avoid. My approach to avoid using anything beyond simple math is to visualize the problem. Do this. Cut a slice through the cork plug from top to bottom, perpendicular to the top ridge. The cross-section looks like this:
The triangle is the cross-section of the cork plug at some arbitrary place through the top ridge. The rectangle is the cross-section of a cylinder the same height and diameter as the plug. The triangle has ½ the area of the rectangle.
The volume of the cork plug is the sum of many thin triangular slices, such as those shown. The volume of the cylinder is the sum of many rectangular slices, such as those shown. The volume of the cork plug is ½ the volume of the cylinder.
And now for the answer from the book. The book is Martin Gardner’s Mathematical Puzzles & Diversions, from 1959. The solution is given on page 58:
4. Any vertical cross section of the cork plug at right angles to the top edge and perpendicular to the base will be a triangle. If the cork were a cylinder of the same height, corresponding cross sections would be rectangles. Each triangular cross section is obviously 1/2 the area of the corresponding rectangular cross section. Since all the triangular sections combine to make up the cylinder, the plug must be 1/2 the volume of the cylinder. The cylinder’s volume is 2π, so our answer is simply π. (This solution is given in “No Calculus, Please,” by J. H. Butchart and Leo Moser in Scripta Mathematica, September-December 1952.)
Actually, the process used to compute this volume is what formed the basis for Newton’s integral calculus. However, since no math formula is involved, we get away with saying we didn’t use calculus.
This week’s Quiz Question will be posted tomorrow at 5 a.m. It might be a bit harder. Then maybe not.