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24 Mar 2020, 07:55
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A cube with an edge length of 6 contains the largest possible sphere that completely fits inside of it. If the volume V of a sphere with radius r is given by the formula \(V = \frac{4}{3}\pi r^3\), what is the volume of the empty space inside the cube?
A. 36(π^3 − 6)
B. 36π
C. 36(6 − π)
D. 36(π − 1)
E. 36(4 − π)
A. 36(π^3 − 6)
B. 36π
C. 36(6 − π)
D. 36(π − 1)
E. 36(4 − π)
Archit3110
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24 Mar 2020, 10:21
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Bunuel
since largest sphere is fitted inside cube of 6 cm ; then its dia = 6cm or radius = 3cm
area of sphere = 4/3 * 27*pi ; 36 pi
and area of cube ; 6^3 ; 216
so empty space = 216-36pi
IMO C; 36(6 − π)
24 Mar 2020, 10:43
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A cube with an edge length of 6 contains the largest possible sphere that completely fits inside of it. If the volume V of a sphere with radius r is given by the formula \(V = \frac{4}{3}\pi r^3\), what is the volume of the empty space inside the cube?
A. 36(π^3 − 6)
B. 36π
C. 36(6 − π) --> correct: the largest possible sphere of radius "r" that completely fits inside of the cube of side "a", so 2r =a =6 => r=3. Empty space inside cube = cube volume - sphere volume = \( a^3- \frac{4}{3}\pi r^3\) = \( 6^3- \frac{4}{3}\pi *3^3\) = \( 6^3- 6^2\pi \) = \( 36(6- \pi) \)
D. 36(π − 1)
E. 36(4 − π)
A. 36(π^3 − 6)
B. 36π
C. 36(6 − π) --> correct: the largest possible sphere of radius "r" that completely fits inside of the cube of side "a", so 2r =a =6 => r=3. Empty space inside cube = cube volume - sphere volume = \( a^3- \frac{4}{3}\pi r^3\) = \( 6^3- \frac{4}{3}\pi *3^3\) = \( 6^3- 6^2\pi \) = \( 36(6- \pi) \)
D. 36(π − 1)
E. 36(4 − π)
