Answer to Question #193517 in Geometry for Lauren Palmer

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.

Answer 1

First of all, I should calculate the volume of the hourglass flask. The exact dimensions are not given, therefore I should express them literally. The shape of the flask is close to a cone. The volume of the cone is equal to one third of the product of the area of ​​the base and the height.

$$V=S*h/3=hπr^2/3$$

.

And than I can calculate the time for the turn.The speed is equal to the volume of sand per unit of time, therefore

$$v=V/t => t=V/v;$$

where v is velocity, V - volume, t - time.

And if express it with the flask size:

$$t=hπr^2/3v=hπr^2 m*m^2/(3*16 mm^3/s)$$

$$t=hπr^2/3v=hπr^2 m*m^2 *10^9/(3*16 m^3/s)$$ $$t=hπr^2/48*10^9 seconds.$$