The hemisphere problem referred to here was described in the previous post. To reword it somewhat, consider this hemisphere, half of a sphere of radius r. The orange cross-section is a circle parallel to the hemisphere’s yellow, circular base.
We are to find the height of the yellow section with the orange circular top (which I shall call x), as a fraction of r, such that the yellow and red sections above have equal volumes.
Since the volume of a hemisphere is (2/3)πr³ (that’s half a sphere’s volume), each of these two sections must have half the hemisphere’s volume, or (1/3)πr³.
Moreover, the top (red) portion is a “spherical cap,” described here on Wikipedia, as was pointed out to me, by a friend, on Facebook. On that Wikipedia page, you can find this diagram, as well as the formula shown below for the volume of the purple spherical cap in the diagram.
Now, as our goal is to find x, as described at the top of this post, it important to remember that r = x + h, where r is the radius of the original sphere (and height of the hemisphere), h is the height of the spherical cap, and x is the height of the hemisphere, after the spherical cap is removed. We now have two expressions for the volume of the spherical cap: (1/3)πr³ (because it is a fourth of the volume of the original sphere), and (1/6)πh(3a² + h²) from the Wikipedia article on the spherical cap (so all of this assumes, then, accuracy in that Wikipedia article). Setting them equal to each other,
(1/3)πr³ = (1/6)πh(3a² + h²)
Next, I’ll clean this up by multiplying left and right by 6/π, to cancel fractions and π from both sides.
2r³ = h(3a² + h²)
A right triangle exists in the blue-and-purple figure above, and the unlabeled leg is x, the problem’s original goal. I’ll add an “x” to this diagram.
Using this right triangle, and the Pythagorean Theorem, it can be seen that a² = r² – x². Also, since r = x + h, it follows that h = r – x. By substitution for a and h, then,
2r³ = h(3a² + h²)
2r³ = (r – x)[3(r² – x²) + (r – x)²], which then distributes to
2r³ = (r – x)(3r² – 3x² + r² – 2rx + x²), which expands as
2r³ = 3r³ – 3rx² + r³ – 2r²x + rx² -3r²x + 3x³ – r²x + 2rx² – x³, which simplifies to
0 = 2r³ – 6r²x + 2x³, which becomes, by division:
0 = r³ – 3r²x + x³.
I’m trying to find x, as a fraction of r, meaning that x = kr, and I want k. On that basis, I’ll now substitute kr for each x in the last equation above.
0 = r³ – 3r²(kr) + (kr)³, which then becomes
0 = r³ – 3kr³ + k³r³, or
3kr³ = r³ + k³r³, and then dividing by r³ yields
3k = 1 + k³, which can be rearranged to
3k – k³ = 1, which factors,on the left side, to yield
(k)(3 – k²) = 1.
For k and (3 – k²) to have a product of one, they must be reciprocals. Therefore, 1/k = 3 – k². I can then graph y = 1/k, as well as y = 3 – k², and find the solution by seeing where the graphed functions intersect above the k-axis, with a “k” value between zero and one, since no value of k less than zero or greater than one would make sense, as a solution to the original problem. (I’ll be using k coordinates, and a k-axis, in place of the usual x coordinates and x-axis.) Here’s the initial graph:
The only intersection in the specified range of zero to one is between point A and point B, so I brought them together, as closely as I could get Geometer’s Sketchpad to let me.
With points A and B almost on top of each other, k = 0.34958 by one equation, and k = 0.34820 by the other. To two significant figures, then, I can conclude that the horizontal cut in the original problem should be made 35% of the way from the base of the hemisphere to the hemisphere’s top.
Using a graphing calculator, a more precise answer of 0.34729636 was obtained. I’d still like to have an exact answer, but this will do for now.
Later addition: a helpful reader led me to a Wolfram Alpha site where I could get an exact answer, as well as a decimal approximation with a greater degree of precision. In the pic below, I have omitted the two solutions of the third-order polynomial which are not the one solution of interest. Here’s the one which is:
Now, however, I have another mystery: how can an exact answer, with all those imaginary units in it, have a real-only approximation? To this question, at least for now, I don’t even have the beginnings of an answer.
Even later post-script: I have been assured by friends on Facebook that the imaginary units in the above exact solution somehow cancel, although I must concede that I still do not see how, myself. I’ve also been shown another way to express the solution, for 2sin(π/18) is also ≈ 0.3472963553338606977034333. This surprised me, due the the lack of any explicit appearance by a π/18 (= 10°) angle in the original problem, and the fact that no trigonometric functions were used to solve it.
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I assumed a radius of 1 since we only need to know the fraction. So we also know that half a hemisphere is 1/3pi. Integrating slices from 0 to h (height) of a slice area pi(1-x^2) and equating the result with 1/3pi, in two lines we get the cubic in standard form h^3-3h+1=0.
We can solve cubics in standard form using trig.
If we let h=ucos(t) where u=2, we get 8cos^3(t)-6cos(t)+1=0.
Divide by 2 (actually 4/u^3) and we have 4cos^3(t)-3cos(t)+1/2=0. (1)
Since cos(3t)=4cos^3(t)-3cos(t), we can rearrange as 4cos^3(t)-3cos(t)-cos(3t)=0 (2)
Now equate -cos(3t)=1/2 (by comparing (1) and (2) and we get t=arccos(-1/2)/3
since h=ucos(t) then h=2cos(arccos(-1/2)/3)
there are three solutions for this but only 1 that is less than 1:
So the answer is 2sin(10) ot 2cos(80).
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It’s a fun calculus exercise. 🙂
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