Tennis balls with a diameter of 2.5 inches are packaged 3 to a can. The can is a cylinder.
Find the volume of the space in the can that is not occupied by tennis balls. Assume that the balls touch the sides, top, and bottom of the can.
Round your answer to the nearest hundredth.
What is the volume not occupied by the balls?
Answer (3)
a) First we calculate the volume of each ball:
V = 3 4 π r 3 V = 3 4 ∗ ( 3 , 14 ) ∗ ( 1.25 ) 3 = 8 , 177 s q u a re in c h
b) Now we calculate the volume of tree balls:
V T = 3 ∗ 8 , 177 = 24.531 s q u a re in c h
c) Now we calculate the volume of cylinder:
c1) Base area:
A b = π r 2 A b = 3 , 14 ∗ ( 1 , 25 ) 2 = 4.906 s q u a re in c h
c2) Cylinder height
h = 3 ∗ ( 2.5 ) = 7.5 in c h
c3) Cylinder Volume:
V C = A b ∗ h V C = 4.906 ∗ 7.5 = 36.795 c u bi c in c h
d) Finally we calculate the internal space:
s = 36.795 − 24.531 = 12 , 264 c u bi c in c h
d = 2.5 in r = 2 d = 2 2.5 = 1.25 in π = 3.14 V o l u m e o f a C y l in d er : V = π r 2 h
h = 3 ⋅ d = 3 ∗ 2.5 = 7.5 in V c = 3.14 ⋅ ( 1.25 ) 2 ⋅ 7.5 = 23.55 ∗ 1.5625 = 36.80 i n 3
T h e v o l u m e o f t ree ba ll s : V 3 b = 3 ⋅ 3 4 π r 3 = 4 π r 3 V 3 b = 4 ⋅ 3.14 ⋅ ( 1.25 ) 3 = 12.56 ⋅ 1.9531 ≈ 24.53 i n 3 V c − V 3 b = 36.80 − 24.53 = 12.27 i n 3 A n s w er : T h e v o l u m e n o t occ u p i e d b y ba ll s i t 12.27 i n 3
The volume of space in the can not occupied by the three tennis balls is approximately 12.29 cubic inches. This calculation involves finding the volume of both the cylinder can and the total volume of the spheres (tennis balls) within it. After computing both volumes, the empty space is found by subtracting the volume of the tennis balls from the volume of the can.
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