Nested Spheres and n-gonal Prisms

Level pending

Let n > = 4 n >= 4 be a fixed positive integer and consider an n n -gonal prism whose bases are regular n n gons and faces are congruent rectangles.

For each positive integer m m ,

Let V c ( m ) V_{c}(m) be the volume of the largest n n -gonal prism that can be inscribed in a sphere of volume V s ( m ) V_{s}(m)

Let V s ( m + 1 ) V_{s}(m + 1) be the volume of the largest sphere that be inscribed in the n n -gonal prism of volume V c ( m ) V_{c}(m) .

Let V s ∗ ( n ) = ∑ m = 1 ∞ V s ( m ) V^{*}_{s}(n) = \sum_{m = 1}^{\infty} V_{s}(m) and V c ∗ ( n ) = ∑ m = 1 ∞ V c ( m ) V^{*}_{c}(n) = \sum_{m = 1}^{\infty} V_{c}(m) and let the volume of the initial sphere V s ( 1 ) = 1 V_{s}(1) = 1 .

Find: lim ⁡ n → ∞ V s ∗ ( n ) ∗ V c ∗ ( n ) \lim_{n \rightarrow \infty} V^{*}_{s}(n) * V^{*}_{c}(n) , and express the result to five decimal places.

Note: My intention is not to reduce the problem to nested spheres and right circular cylinders, but you will get the correct result if you do so., since lim ⁡ n → ∞ V c ( n ) = V c y l i n d e r \lim_{n \rightarrow \infty} V_{c}(n) = V_{cylinder} , where V c ( n ) V_{c}(n) is the volume of a n n -gonal prism.


The answer is 2.78061.

Rocco Dalto by Rocco Dalto , 67, USA

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1 solution

Rocco Dalto
Dec 9, 2017

For area of n − g o n n - gon :

Let B C = x BC = x be a side of the n − g o n n - gon , A C = A B = r ∗ AC = AB= r^{*} , A D = h ∗ AD = h^* , and ∠ B A D = π n \angle{BAD} = \dfrac{\pi}{n} .

x 2 = r ∗ sin ⁡ ( π n ) ⟹ r ∗ = x 2 sin ⁡ ( π n ) ⟹ h ∗ = r ∗ cos ⁡ ( π n ) ⟹ \dfrac{x}{2} = r^{*} \sin(\dfrac{\pi}{n}) \implies r^{*} = \dfrac{x}{2 \sin(\dfrac{\pi}{n})} \implies h^{*} = r^{*} \cos(\dfrac{\pi}{n}) \implies

A n − g o n = n 2 sin ⁡ ( 2 π n ) r ∗ 2 ⟹ A_{n - gon} = \dfrac{n}{2} \sin(\dfrac{2\pi}{n}) {r^{*}}^2 \implies The volume of the n n -gonal prism is V c = n 2 sin ⁡ ( 2 π n ) r ∗ 2 H V_{c} = \dfrac{n}{2} \sin(\dfrac{2\pi}{n}) {r^{*}}^2 H

Let H H be the height of the given n − g o n a l n-gonal prism.

In the right triangle above: A C = H 2 , B C = r ∗ , A B = R ⟹ AC = \dfrac{H}{2}, BC = r^{*}, AB = R \implies

r ∗ 2 = 4 R 2 − H 2 4 ⟹ V p ( H ) = n 8 sin ⁡ ( 2 π n ) ( 4 R 2 H − H 3 ) ⟹ d V c d H = n 8 sin ⁡ ( 2 π n ) ( 4 R 2 − 3 H 2 ) = 0 ⟹ H = 2 R 3 ⟹ {r^{*}}^2 = \dfrac{4 R^2 - H^2}{4} \implies V_{p}(H) = \dfrac{n}{8} \sin(\dfrac{2\pi}{n}) (4R^2 H - H^3) \implies \dfrac{dV_{c}}{dH} = \dfrac{n}{8} \sin(\dfrac{2\pi}{n}) (4 R^2 - 3 H^2) = 0 \implies H = \dfrac{2R}{\sqrt{3}} \implies

r ∗ 2 = 2 3 R 2 ⟹ r ∗ = 2 3 R ⟹ V c ( 1 ) = n 2 3 π sin ⁡ ( 2 π n ) V s ( 1 ) {r^{*}}^2 = \dfrac{2}{3} R^2 \implies r^{*} = \sqrt{\dfrac{2}{3}} R \implies V_{c}(1) = \dfrac{n}{2\sqrt{3}\pi} \sin(\dfrac{2\pi}{n}) V_{s}(1)

For the inscribed sphere:

The radius of the inscribed sphere r = h ∗ = r ∗ cos ⁡ ( π n ) = 2 3 cos ⁡ ( π n ) R ⟹ V s ( 2 ) = 2 3 2 3 cos ⁡ 3 ( π n ) V s ( 1 ) r = h^{*} = r^{*} \cos(\dfrac{\pi}{n}) = \sqrt{\dfrac{2}{3}} \cos(\dfrac{\pi}{n}) R \implies V_{s}(2) = \dfrac{2}{3} \sqrt{\dfrac{2}{3}} \cos^{3}(\dfrac{\pi}{n}) V_{s}(1)

Let r 1 = R , r 2 = r = r ∗ cos ⁡ ( π n ) = 2 3 cos ⁡ ( π n ) R , r 1 ∗ = 2 3 R , H 1 = 2 3 R ⟹ r_{1} = R, \:\ r_{2} = r = r^{*} \cos(\dfrac{\pi}{n}) = \sqrt{\dfrac{2}{3}} \cos(\dfrac{\pi}{n}) R, \:\ r_{1}^{*} = \sqrt{\dfrac{2}{3}} R, \:\ H_{1} = \dfrac{2}{\sqrt{3}} R \implies

r 2 ∗ = 2 3 r 2 = 2 3 ( 2 3 cos ⁡ ( π n ) ) R r_{2}^{*} = \sqrt{\dfrac{2}{3}} r^2 = \sqrt{\dfrac{2}{3}} (\sqrt{\dfrac{2}{3}} \cos(\dfrac{\pi}{n})) R

H 2 = 2 3 ( 2 3 cos ⁡ ( π n ) ) R H^{2} = \dfrac{2}{\sqrt{3}} (\sqrt{\dfrac{2}{3}} \cos(\dfrac{\pi}{n})) R

r 3 = ( 2 3 cos ⁡ ( π n ) ) 2 R r_{3} = (\sqrt{\dfrac{2}{3}} \cos(\dfrac{\pi}{n}))^2 R

r 3 ∗ = 2 3 ( 2 3 cos ⁡ ( π n ) ) ) 2 R r_{3}^{*} = \sqrt{\dfrac{2}{3}} (\sqrt{\dfrac{2}{3}} \cos(\dfrac{\pi}{n})))^2 R

H 3 = 2 3 ( 2 3 cos ⁡ ( π n ) ) 2 R H_{3} = \dfrac{2}{\sqrt{3}} (\sqrt{\dfrac{2}{3}} \cos(\dfrac{\pi}{n}))^2 R

In General:

r ( m ) = ( 2 3 cos ⁡ ( π n ) ) m − 1 R r(m) = (\sqrt{\dfrac{2}{3}} \cos(\dfrac{\pi}{n}))^{m - 1} R

r ∗ ( m ) = 2 3 ( 2 3 cos ⁡ ( π n ) ) m − 1 R r^{*}(m) = \sqrt{\dfrac{2}{3}} (\sqrt{\dfrac{2}{3}} \cos(\dfrac{\pi}{n}))^{m - 1} R

H ( m ) = 2 3 ( 2 3 cos ⁡ ( π n ) ) m − 1 R H(m) = \dfrac{2}{\sqrt{3}} (\sqrt{\dfrac{2}{3}} \cos(\dfrac{\pi}{n}))^{m - 1} R

⟹ V c ( m ) = n 2 3 π sin ⁡ ( 2 π n ) ( 2 3 cos ⁡ ( π n ) ) 3 ( m − 1 ) ∗ V s ( 1 ) \implies V_{c}(m) = \dfrac{n}{2\sqrt{3} \pi} \sin(\dfrac{2\pi}{n}) (\dfrac{2}{3} \cos(\dfrac{\pi}{n}))^{3(m - 1)} * V_{s}(1) and V s ( m ) = ( 2 3 cos ⁡ ( π n ) ) 3 ( m − 1 ) ∗ V s ( 1 ) ⟹ V_{s}(m) = (\dfrac{2}{3} \cos(\dfrac{\pi}{n}))^{3(m - 1)} * V_{s}(1) \implies

V c ∗ ( n ) = ∑ m = 1 ∞ V c ( m ) = n 2 3 π sin ⁡ ( 2 π n ) ( 1 1 − ( 2 3 cos ⁡ ( π n ) ) 3 ) V^{*}_{c}(n) = \sum_{m = 1}^{\infty} V_{c}(m) = \dfrac{n}{2\sqrt{3}\pi} \sin(\dfrac{2\pi}{n}) (\dfrac{1}{1 - (\sqrt{\dfrac{2}{3}} \cos(\dfrac{\pi}{n}))^3}) and V s ∗ ( n ) = ( 1 1 − ( 2 3 cos ⁡ ( π n ) ) 3 ) V^{*}_{s}(n) = (\dfrac{1}{1 - (\sqrt{\dfrac{2}{3}} \cos(\dfrac{\pi}{n}))^3})

Let j ( n ) = n 2 3 π sin ⁡ ( 2 π n ) j(n) = \dfrac{n}{2\sqrt{3}\pi} \sin(\dfrac{2\pi}{n})

Using the inequality: cos ⁡ ( x ) < sin ⁡ ( x ) x < 1 ⟹ \cos(x) < \dfrac{\sin(x)}{x} < 1 \implies

cos ⁡ ( 2 π n ) < n 2 π sin ⁡ ( 2 π n ) < 1 ⟹ 1 3 cos ⁡ ( 2 π n ) < j ( n ) < 1 3 ⟹ lim ⁡ n → ∞ j ( n ) = 1 3 \cos(\dfrac{2\pi}{n}) < \dfrac{n}{2\pi} \sin(\dfrac{2\pi}{n}) < 1 \implies \dfrac{1}{\sqrt{3}} \cos(\dfrac{2\pi}{n}) < j(n) < \dfrac{1}{\sqrt{3}} \implies \lim_{n \rightarrow \infty} j(n) = \dfrac{1}{\sqrt{3}}

⟹ lim ⁡ n → ∞ V c ∗ ( n ) \implies \lim_{n \rightarrow \infty} V^{*}_{c}(n) = 3 ∗ V s ( 1 ) 3 3 − 2 2 \dfrac{3 * V_{s}(1)}{3\sqrt{3} - 2\sqrt{2}} and lim ⁡ n → ∞ V s ∗ ( n ) \lim_{n \rightarrow \infty} V^{*}_{s}(n) = 3 3 ∗ V s ( 1 ) 3 3 − 2 2 ⟹ \dfrac{3 \sqrt{3} * V_{s}(1)}{3\sqrt{3} - 2\sqrt{2}} \implies

lim ⁡ n → ∞ V c ∗ ( n ) ∗ V s ∗ ( n ) \lim_{n \rightarrow \infty} V^{*}_{c}(n) * V^{*}_{s}(n) = 9 3 ∗ V s ( 1 ) 2 ( 3 3 − 2 2 ) 2 \dfrac{9\sqrt{3} * V_{s}(1)^2}{(3 \sqrt{3} - 2\sqrt{2})^2}

Letting V s ( 1 ) = 1 ⟹ lim ⁡ n → ∞ V c ∗ ( n ) ∗ V s ∗ ( n ) V_{s}(1) = 1 \implies \lim_{n \rightarrow \infty} V^{*}_{c}(n) * V^{*}_{s}(n) = 9 3 ( 3 3 − 2 2 ) 2 = 2.78061 \dfrac{9\sqrt{3}}{(3\sqrt{3} - 2\sqrt{2})^2} = \boxed{2.78061}

Using right circular cones:

From above the volume of the n n -gonal prism is V c ( n ) = n 2 sin ⁡ ( 2 π n ) r ∗ 2 H V_{c}(n) = \dfrac{n}{2} \sin(\dfrac{2\pi}{n}) {r^{*}}^2 H

Using the inequality cos ⁡ ( x ) < sin ⁡ ( x ) x < 1 ⟹ cos ⁡ ( 2 π n ) < n 2 π ∗ sin ⁡ ( 2 π n ) < 1 ⟹ π ∗ cos ⁡ ( 2 π n ) < n 2 sin ⁡ ( 2 π n ) < π \cos(x) < \dfrac{\sin(x)}{x} < 1 \implies \cos(\dfrac{2\pi}{n})< \dfrac{n}{2\pi} * \sin(\dfrac{2\pi}{n}) < 1 \implies \pi * \cos(\dfrac{2\pi}{n}) < \dfrac{n}{2} \sin(\dfrac{2\pi}{n}) < \pi

⟹ lim ⁡ n → ∞ V c ( n ) = π r ∗ 2 H = V c y l i n d e r \implies \lim_{n \rightarrow \infty} V_{c}(n) = \boxed{\pi {r^{*}}^2 H = V_{cylinder}}

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