The formula for the volume of a sphere is V=(4/3)πr³. We will only need half of this, so our formula is V=(1/2)(4/3)πr³=(4/6)πr³=(2/3)πr³. Since the diameter of the sink is 20 in, the radius is half of that or 10 in. Substituting that in, we have:
V=(2/3)π(10³)=(2/3)(1000)π=2000π/3 in³.
The formula for the volume of the conical cup is V=(1/3)πr²h. Our radius is 1/2 of the 8 in diameter, or 4 in. Using our information we have V=(1/3)π(4²)(6)=(1/3)π(96)=32π in³.
To find out how many cups it will take to empty the sink, we divide the volume of the sink by the volume of the cup:
[tex]\frac{2000\pi}{3} \div 32\pi
\\
\\=\frac{2000\pi}{3} \div \frac{32\pi}{1}
\\
\\=\frac{2000\pi}{3} * \frac{1}{32\pi}
\\
\\=\frac{2000\pi *1}{3*32\pi}
\\
\\=\frac{2000*1}{3*32}=\frac{2000}{96} \approx 21[/tex]
The volume of the cylindrical cup is given by the formula V=πr²h. Our radius is half of the diameter of 4, or 2 in. Using our information we have V=π(2²)(6)=24π in³. To determine how many cups it would take to empty the sink we divide the volume of the sink by the volume of the cup:
[tex]\frac{2000\pi}{3} \div 24\pi
\\
\\=\frac{2000\pi}{3} \div \frac{24\pi}{1}
\\
\\=\frac{2000\pi}{3} * \frac{1}{24\pi}
\\
\\=\frac{2000\pi *1}{3*24\pi}=\frac{2000*1}{3*24}=\frac{2000}{72} \approx 28[/tex]
