General Idea:
The volume of cylinder is given by [tex] \pi r^{2} h [/tex], 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 [tex] =\pi r^{2} h=\pi* 8^{2} *9=576\pi cm^{3} [/tex]
Volume of half cylinder [tex] =\frac{576\pi }{2} =288\pi cm^{3} [/tex]
Step 2: Let x be the number of minutes of filling the sand.
[tex] 6 cm^{3} [/tex] of sand filled every 15 seconds, there are four 15 seconds in a minute.
So volume of sand filled in 1 minute[tex] = 6*4 cm^{3} = 24 cm^{3} [/tex].
[tex] 8 cm^{3} [/tex] 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[tex] =24 cm^{3} - 8cm^{3}=16cm^{3} [/tex]
Volume of sand filled in x minutes [tex] = 16x [/tex].
We need to set up an equation to find the number of minutes need to fill half the volume in cylindrical vase.
[tex] 16x = 288\pi \\ \\ x=\frac{288\pi}{16} \\ \\ x=18\pi \\ \\ x=57 minutes [/tex]
Conclusion:
The number of minutes required for the base be half filled with sand is 57
