A chocolate company makes chocolate malt balls that are 0.75 inches in diameter. the carton they are to be packed in is approximately a rectangular prism with the dimensions 3 inches by 3 inches by 7.5 inches. how many malt balls will fit in the carton?

(reminder: you can assume that for random packing space 190% of the volume of the sphere is needed to pack the spheres and volume if sphere is 4/3π x r^3)

A. 160 balls

B. 210 balls

C. 305 balls

D. 82 balls

A. 160 balls

Step-by-step explanation:

First we find the volume of each sphere.

diameter = d = 0.75 in.

radius = r = d/2 = 0.75/2 in. = 0.375 in.

V = (4/3) π (r^3) = (4/3) (3.14159) (0.375 in.) ^3 = 0.221 in.^3

Because of the assumption about the space occupied by a sphere in random packing, we now calculate 190% of the volume of the sphere.

190%V = 1.9 * 0.221 in.^3 = 0.420 in.^3

Now we calculate the volume of the box.

V = LWH = 3 in. * 3 in. * 7.5 in. = 67.5 in.^3

To find out the number of balls that fit in the box, we divide the volume of the box by the randomly packed ball volume.

number of balls = (67.5 in.^3) / (0.420 in.^3) = 160.8

Answer: 160 balls