Question

Water flows at $12\text{inches/sec}$ through a cylindrical pipe with a diameter of $10\text{inches}$. A cylindrical tank with a diameter of 60 inches and height of $120\text{inches}$ collects the water. How many minutes will it take to fill $75\%$ of the tank?
[Hint: Calculate the volume of water coming out of pipe every second]

Answer 1

Detailed step-by-step solution:
Diameter of the pipe, $d=10\text{inches}$
Radius of the pipe, $r=\frac{d}{2}=\frac{10}{2}=5\text{inches}$
The area of the cross section of the pipe = $\pi r^2$
Speed of water = $12\text{inches/sec}$

Diameter of the cylindrical tank, $D=60\text{inches}$
Radius of the cylindrical tank, $R=\frac{D}{2}=\frac{60}{2}=30\text{inches}$

Height of the cylindrical tank, $H=120\text{inches}$

Calculating the rate of coming out of water from the pipe:
The volume of the water flowing through the pipe in $1\text{sec}=\pi r^2\times 12$
$=\pi \times 5^2\times 12$
$=300\pi \text{cubic inches}$

The volume of the water coming out of the pipe every sec is equal to the volume of the water collected by the tank every sec.
The volume of the water collected every $\text{sec}=300\pi \text{cubic inches}$
I.e.,
Volume flow rate =$300\pi \text{cubic inches/sec}=300\pi \text{cubic inches}$ (volume flow rate = volume of water collected by the tank every sec)

Calculating the volume of the tank:
Volume of the cylindrical tank = $\pi R^{2}H$ (where $R$ is the radius of the base and $H$ is the height)
$=\pi \times {30}^2\times 120$
$=\pi \times 30\times 30\times 120$
$=10,800\pi \text{cubic inches}$

$75\%$ of the volume of the tank =$75\%\times 10,800\pi \text{cubic inches}$

$=\frac{75}{100}\times 10,800\pi \text{cubic inches}$
$=81,000\pi \text{cubic inches}$
The time taken to fill $75\%$ of the tank will be calculated by dividing $75\%$ of the volume of the tank by the volume flow rate.
I.e.,
Time taken = $\frac{81000\pi \text{cubic inches}}{300\pi \text{cubic inches/sec}}$

$=270\text{sec}$

$=\frac{270}{60}\text{min}$ $(1\text{min}=60\text{sec},1\text{sec}=\frac{1}{60}\text{min})$

$=4.5\text{min}$

It will take $4.5\text{min}$ to fill $75\%$ of the tank
.
➡Option C is correct.