Question

Imagine you're playing a board game that involves an hourglass filled with sand. Once

all of the sand falls to the bottom, your turn is up and it's the next player's turn. If the
sand falls at a rate of 16 cubic millimeters per second, how much time do you have for
your turn? It may be helpful to first calculate the volume of the sand and then calculate
the time.
8 mm
12 mm

Answer 1

To calculate the volume of sand in the hourglass, we need to use the formula for the volume of a cone, which is:

V = (1/3)πr^2h

where V is the volume, r is the radius of the base of the cone, and h is the height of the cone.

In this case, the hourglass can be thought of as two cones joined at their bases. The radius of each base is 4 mm (half of the width of the hourglass), and the height of each cone is 12 mm (the full height of the hourglass). So the volume of sand in the hourglass is:

V = 2 × (1/3)π(4 mm)^2(12 mm) = 128π mm^3

Now we can calculate the time it takes for the sand to fall to the bottom of the hourglass. We know that the rate of sand falling is 16 cubic millimeters per second, so we can use the formula:

V = rt

where r is the rate of sand falling and t is the time it takes for the sand to fall. Rearranging this formula, we get:

t = V/r = (128π mm^3)/(16 mm^3/s) = 8π s

So you have 8π seconds, or approximately 25.13 seconds, for your turn.