Problems Plus 8: Volume of a Solid from its Shadows

about 1 month ago

In this video I obtain the volume of the largest solid whose shadows are a square of height 1, a circle, and an isosceles triangle. I first start off with a circular cylinder and then cut off the sides to form a triangle when viewed from the side. I calculate the volume by taking thin vertical slices and summing up using an integral. The volume turns out to be about 0.45. Lastly, I show that the volume can be made arbitrarily small as long as the skeleton is maintained.

Timestamps:

Problem 8: Volume of a solid from its shadows: 0:00
- Review of Exercise 44: Solid with square, isosceles triangle, and circle shadows: 0:44
Solution to (a): Volume of the largest such solid: 1:51
Start with a circular cylinder and shave off excess: 3:05
Cut solid with planes a parallel to the x-axis and form a triangle: 6:02
Compute volume of the solid via vertical slices parallel to xz plane: 8:21
- Base of the solid is a circle with radius 0.5: 9:15
- Obtain height from a triangle: 13:57
- Determine cross section area: 15:54
Solve for the volume by taking an integral: 18:51
- First integral is a quarter-circle: 21:10
- Second integral is solved by substitution: 22:38
Volume is approximately 0.45: 28:50
- Calculation check using an integral calculator: 30:39
Solution to (b): There is no smallest volume: 32:56
- Can have arbitrarily small volume as long as the skeleton is still in tact: 34:33