Question

An empty cylindrical vase with a radius of 8 cm and a height of 9 cm is filled with 6 cm^3 of sand every 15 seconds. At the same time 8cm^3 of sand is taken out of the vase every minute. After how many minutes will the base be half filled with sand?

Answer 1

General Idea:

The volume of cylinder is given by $\pi r^{2} h$, where r is the radius and h is the height of the cylinder.

Applying the concept:

Step 1: We need to find the volume of full cylinder with the given dimensions using the formula.

Volume of full cylinder $=\pi r^{2} h=\pi* 8^{2} *9=576\pi cm^{3}$

Volume of half cylinder $=\frac{576\pi }{2} =288\pi cm^{3}$

Step 2: Let x be the number of minutes of filling the sand.

$6 cm^{3}$ of sand filled every 15 seconds, there are four 15 seconds in a minute.

So volume of sand filled in 1 minute$= 6*4 cm^{3} = 24 cm^{3}$.

$8 cm^{3}$ of sand taken out of cylindrical vase every minute.

Net volume of sand filled in 1 minute = Volume of sand filled in the vase in one minute - Volume of sand taken out in 1 minute

Net volume of sand filled in 1 minute$=24 cm^{3} - 8cm^{3}=16cm^{3}$

Volume of sand filled in x minutes $= 16x$.

We need to set up an equation to find the number of minutes need to fill half the volume in cylindrical vase.

$16x = 288\pi \\ \\ x=\frac{288\pi}{16} \\ \\ x=18\pi \\ \\ x=57 minutes$

Conclusion:

The number of minutes required for the base be half filled with sand is 57