Question

"A gumball machine is in the shape of a sphere with a radius of 6 inches. A store manager wants to fill up the machine with miniature gumballs, which have a radius of 1/3in. How many gumballs will fit in the machine?"

a. 14.2
b. 5832
c. 1642
d. 972

Answer 1

So Volume (V) of a sphere is:

`V = (4)/(3) \pi {r}^(3)`
so essentially we need to divide the machine's Volume (V) the gumball volume (v)

`V = (4)/(3) \pi {r}^(3) = (4)/(3) \pi {(6)}^(3) \\ V = (4)/(3)(216) \pi = 288\pi`

`v = (4)/(3) \pi {( (1)/(3) )}^(3) = (4)/(3) * (1)/(27) \pi \\ v = (4)/(81) \pi`
now we divide V by v:

`V/v = (288\pi)/( (4\pi)/(81) ) = (288)/(1) * (81)/(4) = 5832`
so b) 5,832 gumballs can fit in the machine