A sphere is to be formed by melting smaller sphere cubes of diameter 20 mm. The sphere needs to be placed inside a cuboi

VJesus12 wrote: ↑

Sun Dec 27, 2020 7:52 am

A sphere is to be formed by melting smaller sphere cubes of diameter 20 mm. The sphere needs to be placed inside a cuboid of length 60 cm, width 50 cm, and height 40 cm such that it covers the maximum possible volume. How many smaller cubes are needed?

A. 8
B. 80
C. 800
D. 8000
E. 80000

Answer: D

Solution:

The volume of a smaller sphere is 4/3 x π x 10^3 = 4000π/3 mm^3. The largest sphere that can be placed inside the cuboid must have a diameter no larger than the smallest dimension of the cuboid. Therefore, the largest sphere can have a diameter of only 40 cm and thus, its volume is 4/3 x π x 20^3 = 32000π/3 cm^3. Since 4000π/3 mm^3 = 4π/3 cm^3 (recall that 1 cm^3 = 1000 mm^3), the number of smaller spheres needed is (32000π/3) / (4π/3) = 32000/4 = 8000.

Answer: D