Math mini-contest problem for Day 3 on D.Buzz for April 2021 😎

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Math mini-contest problem for Day 3 on D.Buzz for April 2021 😎

Vision has a hemispherical container with height of 8 inches and contains water. The water surface has a radius of 3 inches. How much water is in the container?



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V = (2/3)(pi)(r^2) = 6pi

The answer is 6 pi square inches or 18.850 square inches.

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(Edited)

Answer for Day 3 Math Problem

8.36 cubic inches🎯

Equivalent answers in other cubic units, in either fractional or decimal format, shall be accepted.
Β 

Solution

The problem is about a spherical cap, which is formed by a plane cutting a sphere. In the original problem, the spherical cap is the water content of the hemisphere.

The volume of a spherical cap given the radius of the base of the cap and the radius of the sphere can be obtained using the following formula.

V = Ο€ * hΒ² * (3r - h) / 3

where

  • r = radius of the sphere (or the height of the hemisphere)
  • h = height of the spherical cap

However, it is important to take note that the height of the spherical cap is not equal to the radius of the sphere (unless otherwise indicated). We still need to get the value of h using the values of r (which is 8 inches) and b (which is 3 inches).

To get the height of the spherical cap, we should use the formula that makes use of the radii of the sphere and the base of the spherical cap as posted here.

h = r - √(r² - b²)

which will then produce

h = 8 - √(8Β² - 3Β²) β‰ˆ 0.5838

That value is nicely displayed on the Desmos graph screenshot below (as inspired by @minus-pi)!

minuspi.png

Using the values r = 8 and h = 0.5838 for V = Ο€ * hΒ² * (3r - h) / 3, we get

  • V = Ο€ * hΒ² * (3r - h) / 3
  • V = Ο€ * 0.5838Β² * (3 * 8 - 0.5838) / 3
  • V β‰ˆ 8.357481828

The volume of the spherical cap, which is the water in the bowl in the original problem, is approximately 8.36 cubic inches. πŸ€“

This example would help! πŸ€”

Winner: @minus-pi πŸ…

1 HIVE has been sent to @minus-pi's Hive account. Additionally, since he was able to point out the error in my solution, I also awarded him with 1 extra HIVE!

  • I can't determine how @jfang003 got his answer, since he provided minimal to no solution! 🀯

Participants: @holovision, @ahmadmanga (@ahmadmangazap), @eturnerx (@eturnerx-dbuzz), @paultactico2, @dkmathstats, and @appukuttan66 πŸ€“
Special mentions: @chrisrice, @jancharlest, and @mehmetfix 🀯

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V = Ο€ * hΒ² * (3r - h) / 3
V = Ο€ * 8Β² * (3 * 3 - 8) / 3

hmm, maybe I am on a wrong track here? This is how I understood it:
h is the height of the cap, which is the height at the point where the radius is 3 => h = 0.584
r is the radius of the sphere, not the cap => r = 8

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I accidentally typed "b" instead of "h" on the formula. I actually thought twice whether to use "b" or "r", because "r" or radius can refer to either the radius of the sphere, or just the radius of the base. πŸ˜…

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I just edited the solution to properly reflect "b" as the radius of the base of the spherical cap. 😁 Thanks for pointing it out! πŸ˜…

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hmm, still not fully convinced, see here. h is not 8 here, it's the height at the point where water radius is 3:
image.png

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I have detected an error in my solution! πŸ™ˆ Thanks a lot for pointing it out! πŸ˜… I am currently typing the correct solution, along with a surprise! πŸ˜ƒ

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