Math Easy Solutions | Problems Plus 8: Volume of a Solid from its Shadows @mes | Uploaded October 2024 | Updated October 2024, 7 minutes 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
Video notes and playlists:
- HIVE notes: peakd.com/hive-128780/@mes/jlgigkdl
- Full video and playlist: youtube.com/playlist?list=PLai3U8-WIK0G_7Jd6ZORHFQMk8jbwmkos
- Vectors and the Geometry of Space playlist: youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ .
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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
Video notes and playlists:
- HIVE notes: peakd.com/hive-128780/@mes/jlgigkdl
- Full video and playlist: youtube.com/playlist?list=PLai3U8-WIK0G_7Jd6ZORHFQMk8jbwmkos
- Vectors and the Geometry of Space playlist: youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ .
------------------------------------------------------
Become a MES Super Fan! youtube.com/channel/UCUUBq1GPBvvGNz7dpgO14Ow/join
DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate
SUBSCRIBE via EMAIL: https://mes.fm/subscribe
MES Links: https://mes.fm/links
MES Truth: https://mes.fm/truth
Official Website: https://MES.fm
Hive: peakd.com/@mes
Email me: contact@mes.fm
Free Calculators: https://mes.fm/calculators
BMI Calculator: https://bmicalculator.mes.fm
Grade Calculator: https://gradecalculator.mes.fm
Mortgage Calculator: https://mortgagecalculator.mes.fm
Percentage Calculator: https://percentagecalculator.mes.fm
Free Online Tools: https://mes.fm/tools
iPhone and Android Apps: https://mes.fm/mobile-apps
