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# A cube of side s is positioned within a sphere such that each of its

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Math Expert
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A cube of side s is positioned within a sphere such that each of its [#permalink]

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10 Apr 2018, 04:07
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A cube of side s is positioned within a sphere such that each of its corners is tangent to the inside of the sphere and such that the two figures do not otherwise interact. What is the volume of the sphere, in terms of s? (Note: the volume of a sphere is equivalent to $$\frac{4}{3} \pi r^3$$.)

A. $$\frac{s^3\pi \sqrt{2}}{3}$$

B. $$2s^3 \pi \sqrt{3}$$

C. $$\frac{8s^3\pi \sqrt{2}}{3}$$

D. $$4s^3 \pi \sqrt{3}$$

E. $$\frac{s^3\pi \sqrt{3}}{2}$$
[Reveal] Spoiler: OA

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A cube of side s is positioned within a sphere such that each of its [#permalink]

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10 Apr 2018, 05:18
Given that the cube has side s, we want to find the diagonal of the cube which would correspond to the diameter d of the sphere.

So the diagonal of a cube is s * sqrt(3) = d. And we know that V = (4/3) * pi * r^3 => V = (4/3) * pi * (s*sqrt(3)^3 /2) = s^3 * pi * (sqrt 3) / 2 (E)
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A cube of side s is positioned within a sphere such that each of its [#permalink]

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10 Apr 2018, 07:47
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Bunuel wrote:
A cube of side s is positioned within a sphere such that each of its corners is tangent to the inside of the sphere and such that the two figures do not otherwise interact. What is the volume of the sphere, in terms of s? (Note: the volume of a sphere is equivalent to $$\frac{4}{3} \pi r^3$$.)

A. $$\frac{s^3\pi \sqrt{2}}{3}$$

B. $$2s^3 \pi \sqrt{3}$$

C. $$\frac{8s^3\pi \sqrt{2}}{3}$$

D. $$4s^3 \pi \sqrt{3}$$

E. $$\frac{s^3\pi \sqrt{3}}{2}$$

The cube's corners are tangent to the inside of the sphere.
No other parts of the 3D figures touch.

Thus the cube's space diagonal (the line between two corners that do not share a face) = the diameter of the sphere

The cube's space diagonal, d, is $$s\sqrt{3}$$

OR: find the space diagonal from a variation of Pythagorean theorem*:

$$s^2 + s^2 + s^2 = d^2$$
$$3s^2 = d^2$$
$$\sqrt{3s^2}=\sqrt{d^2}$$
$$d = s\sqrt{3}$$

Cube's space diagonal = sphere's diameter, and

$$\frac{diameter}{2}$$ = sphere's radius
$$r = \frac{s\sqrt{3}}{2}$$

Volume, V, of sphere: $$\frac{4}{3} \pi r^3$$
$$r = \frac{s\sqrt{3}}{2}$$

$$V=\frac{4}{3} \pi* (\frac{s\sqrt{3}}{2})^3$$

$$V= \frac{4}{3} \pi* \frac{s^3 3\sqrt{3}}{8}$$

$$V=\frac{s^3\pi \sqrt{3}}{2}$$

Space diagonal of any rectangular prism:
$$(l^2 + w^2 + h^2) = d^2$$. Square side $$s = l, w,$$ and $$h$$.

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Re: A cube of side s is positioned within a sphere such that each of its [#permalink]

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10 Apr 2018, 08:09
d= sroot(3) r= sroot3/2 v=(4/3)pi(sroot3/2)^3-->(4/3)(1/8)(3root3s^3)pi = (1/2)root(3)pi*s^3 E
Re: A cube of side s is positioned within a sphere such that each of its   [#permalink] 10 Apr 2018, 08:09
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