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netzer
netzer
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February 9th, 2019 at 10:59:33 AM permalink
Quote: ThatDonGuy

I get what The Wizard gets


Only partly! You agree with him that the ratio of volume to surface area for the optimal cone is 1/6 but you also believe, along with me, that for the optimal cone h/r = √2. The Wizard believes that r = h = √2/2, so for him h/r = 1. He is using a square right triangle with hypotenuse 1.

Quote: ThatDonGuy

What I think is the flaw in Netzer's solution


"The last part of your answer appears to assume that it is always true." What is always true?

"If t is the angle in radians, and the radius is 1, then the circumference of the wedge is 2 PI r * (t / 2 PI) = t r = t."

No math needed! That's the definition of radian measure: the length of the arc the angle subtends divided by the radius.

"Solve for t such that the slant radius = 1." I am using a slant radius of √3 for a base radius of 1. It's a 1 √2, √3 triangle.

I have made a correction in my previous post:

The volume is πr2h/3 = πh/3 and the surface area is πrR = π√3 so the ratio of volume to surface area is √2/3√3 or approximately 1 to 3.67423

I have been going over the Wizard's development. He defines

r = radius of the base of the cone.
h = height of cone
c = circumference of the base of the cone
S = surface area of cone
V = volume of cone


Quotes from the Wizard are in italics.

He does not assign a symbol to the slant height of the cone, which is the radius of the circle from which the cone is cut. I call it R.

So, the area of the slice is π ­ rπ, thus the area of Pac Man is π ­ (π ­ rπ) = rπ
As a reminder the Pac Man shape is the cone flattened out, so S = rπ.


Actually, it is πrR, so everything that follows is tainted. A quick way to see this would be to check the units on the left and the right side of the equation. The left side is in square units but the right side is in linear units. Adding R to the right side makes it in square units.

Area of a cone, excluding the base.

Let r be the radius of the base and R be the distance from the apex to any point on the circumference of the base. Call this the slant height. The length of the circumference of the base will be 2πr.

Now cut the cone along any R line and flatten it out. It becomes a pizza with a piece missing. If it were whole its area would be πR2, however it has only 2πr of its original circumference of 2πR left and its area is reduced proportionally so it becomes

πR2(2πr/2πR) = πrR

It's as simple as that!
Last edited by: netzer on Feb 9, 2019
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ThatDonGuy
ThatDonGuy
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February 9th, 2019 at 12:15:12 PM permalink
You can't just set a value for r and then assume it's true for all values of r.
Remember, V varies as r3, and A varies as r2, so V/A varies as r.

If r = 1, h = sqrt(2), and d = sqrt(3):
V = PI/3 r2 h = PI/3 * sqrt(2)
A = PI r d = PI sqrt(3)
V/A = sqrt(2) / (3 sqrt(3)) = sqrt(6) / 9 = about 1 / 3.67423, which is what you get

If r = 2, h = 2 sqrt(2), and d = 2 sqrt(3):
V = PI/3 r2 h = PI/3 * 4 * 2 sqrt(2) = PI * 8 sqrt(2)/3
A = PI r d = PI * 2 * 2 sqrt(3) = PI * 4 sqrt(3)
V/A = (8 sqrt(2)) / (3 * 4 sqrt(3)) = (2 sqrt(2) sqrt(3)) / (3 * 3) = 2 sqrt(6) / 9

Both are 1 - sqrt(2) - sqrt(3), but the ratios are different.

And I did find one mistake - when I said that h/r= sqrt(2). I crunched the numbers on a spreadsheet for slant length = 1, and the maximum volume/area is 1/6 when r = h = sqrt(2)/2. I did say earlier in the thread that, when the slant length is fixed (which it is), then the ratio is a maximum when r = h.

h / r = sqrt(2) is correct when either the volume or the lateral surface area is fixed, but in this case, it is the slant length that is fixed.
netzer
netzer
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February 9th, 2019 at 12:34:00 PM permalink
Quote: ThatDonGuy

You can't just set a value for r and then assume it's true for all values of r.

Statement 1:

Remember, V varies as r3, and A varies as r2, so V/A varies as r.

Statement 2:

V/A = sqrt(2) / (3 sqrt(3)) = sqrt(6) / 9 = about 1 / 3.67423, which is what you get



I appreciate the amount of thought you are giving this and I'm glad we're coming closer together, but Statement 1 contradicts Statement 2, does it not? Is V/A a function of r or is it a constant?

V varies as r2 and A varies as rR, and since R varies as r, V/R doesn't vary.

While you were writing I added some observations on the Wizard's solution. If you agree he made a mistake we should tell him before he posts it on his puzzle site. Draw straws?
Last edited by: netzer on Feb 9, 2019
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ThatDonGuy
ThatDonGuy
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February 9th, 2019 at 12:48:15 PM permalink
Quote: netzer

Quote: ThatDonGuy

You can't just set a value for r and then assume it's true for all values of r.

Statement 1:

Remember, V varies as r3, and A varies as r2, so V/A varies as r.

Statement 2:

V/A = sqrt(2) / (3 sqrt(3)) = sqrt(6) / 9 = about 1 / 3.67423, which is what you get



I appreciate the amount of thought you are giving this and I'm glad we're coming closer together, but Statement 1 contradicts Statement 2, does it not? Is V/A a function of r or is it a constant?

While you were writing I added some observations on the Wizard's solution. If you agree he made a mistake we should tell him before he posts it on his puzzle site. Draw straws?


You left out "statement 3":
V/A = (8 sqrt(2)) / (3 * 4 sqrt(3)) = (2 sqrt(2) sqrt(3)) / (3 * 3) = 2 sqrt(6) / 9
This is in line with statement 1 - that V/A varies as r, and is not a constant

And I don't see what mistake Wizard made; as I said when I corrected myself, the V/A ratio is a maximum for a given slant length when h = r, which is what he has (and you agreed with this) in his solution.

However, I think there is an error in his solution, as I get PI (2 - sqrt(2)) instead of PI/2 as the angle to cut out of the tortilla.
netzer
netzer
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February 9th, 2019 at 1:22:01 PM permalink
Quote: ThatDonGuy



And I don't see what mistake Wizard made; as I said when I corrected myself, the V/A ratio is a maximum for a given slant length when h = r, which is what he has (and you agreed with this) in his solution.


I agree that that is what he wrote but I do not agree that it is correct. Also, I think the Wizard will see the error immediately.


Quote: ThatDonGuy


However, I think there is an error in his solution, as I get PI (2 - sqrt(2)) instead of PI/2 as the angle to cut out of the tortilla.


That's 1.84030 radians , or 105.4414 degrees. A little wide, I think.
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netzer
netzer
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February 11th, 2019 at 6:07:19 AM permalink
I'd like to see if we can develop a consensus on a formula as a necessary first step to getting the right answer. You don't have to derive it; just look it up in any reference source on geometric figures.

What is the formula for the surface area of a right circular cone, not including the base?
Let S be the surface area, r be radius of the base, and R the slant height: the distance from any point on the perimeter of the base all the way up the side to the apex. The area can be expressed in terms of r and R. You don't need to know the height of the cone.

S = ? Anybody?
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Wizard
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Wizard
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February 11th, 2019 at 10:24:08 AM permalink
Quote: netzer

What is the formula for the surface area of a right circular cone, not including the base?



Solving for the surface area of a cone is surprisingly tricky.

Not including the base, the formula is pi*radius*slant height.


Without using that formula, what is the surface area of a cone, not including the base, for a cone of height of 12 and radius of 5?
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