Answer to Question #113465 in Geometry for Ben

Question

Twenty-five balls, each with diameter 7 cm, are packed in a single layer in a square box. What is the empty space in the box?

Answer 1

First of all we must find the volume of all 25 balls.

Ball is a sphere, so let's write formula for sphere volume:

$V_s=\dfrac{4}{3}*\pi*r^3$ , where r is a radius of the sphere.

Radius is the half of diameter: r = D/2 = 7/2 = 3.5 cm

To keep calculations more accurate let's find volume of 25 balls at one step. To do that we need to multiply previous formula by 25:

$V_{25} =25*\dfrac{4}{3}*\pi*r^3 = 25*\dfrac{4}{3}*3.14*(3.5)^3=$

$=\dfrac{314}{3}*42.875\approx4487.6$ cm³

Now let's find the volume of our box.

Unfortunately, we don't have accurate box dimensions, so let's find them out.

We have 25 balls packed in one layer. Our box is square, so balls must be packed 5 x 5.

Now we can find length and width of the box. 5 balls in each row, so it'll be 5*7 = 35 cm

length = width = 35 cm.

To find volume of the box we should multiply its height, width and length:

V_b = h * w * l

Because of lacking data we should to consider two cases:

1. Height of the box is the same as width and length - our box is a cube.
2. Height of the box is the same as diameter of balls - our box is a cuboid.

Box volume for case 1:

h = l = w = 35 cm

V_b1 = h * l * w = 35³ = 42875 cm³

Box volume for case 2:

h = 7 cm, l = w = 35 cm

V_b2 = 35 * 35 * 7 = 8575 cm³

Now we can find empty box space for both cases:

V_e1 = V_b1 - V₂₅ = 42875 - 4487.6 = 38387.4 cm³

V_e2 = V_b2 - V₂₅ = 8575 - 4487.6 = 4087.4 cm³