The speed of the water tells us that, every single second, an amount of water equal to the volume of a cylinder with a height of 2 feet and a diameter of 8 inches flows out of the pipe.
B
Find the radius of the tank and the volume of water in it after 5 minutes. Use the formula for the volume of a cylinder to calculate the height of the water.
C
Find the volume of 75 % of the tank. Divide this by the flow of water to find the time it would take to fill up the tank to this point.
A
About 1206 cubic inches
B
About 14.2 inches
C
About 19 minutes
Practice makes perfect
We have a cylindrical tank with a pipe leading water to it. The pipe has a diameter of 8 inches. The water flows through the pipe at a speed of 2 feet per second. We want to find the volume of water flowing out of the pipe every second. Let's start by visualizing the situation.
The speed of the water tells us that, every single second, an amount of water equal to the volume of a cylinder with a height of 2 feet and a diameter of 8 inches flows out of the pipe. We can use the formula for the volume of a cylinder to calculate this.
V=π r^2 h
We know the height h and that the radius r is half of the diameter.
Before we can calculate the volume, we need to convert the height to inches so that our measurements are both in the same unit length. We do this by using a conversion factor.
h= 2 ft * 12 in/1 ft= 24 in
Now, we can calculate the volume.
The volume of water flowing out of the pipe is about 1206 cubic inches per second.
We want to find the height of water in the tank after 5 minutes. Let's start by looking at the dimensions of the tank.
To find the height of the water, we will use the formula for the volume of a cylinder. We need to find the radius r and the volume V.
V=π r^2 hThe diameter of the tank is 15 feet. We can calculate the radius by dividing it by 2.
We get the radius in inches by using a conversion factor. We need to do this so that all of our units of measure are the same.
r= 7.5 ft * 12 in/1 ft= 90 in
Now, we want to find the volume of water after 5 minutes. We know from Part A that 1206 cubic inches of water are added to the tank every second. Let's convert the minutes to seconds so that our units match here as well.
t= 5 min * 60 s/1 min= 300 s
Next, we calculate the total volume of water in the tank after 5 minutes.
V= 1206* 300= 361 800 in^3
Finally, we can find the height of the water after 5 minutes.
The height of the water after 5 minutes is about 14.2 inches.
We want to find the time t it would take in minutes to fill 75 % of the tank. We can do this by finding the volume of 75 % of the tank and then dividing by the flow to the tank per minute.
\begin{gathered}
t=\dfrac{V_\text{tank}}{V_\text{flow}}
\end{gathered}
Let's begin by looking at the dimensions of the tank to find the volume of it.
In Part B, we calculated that the radius is 90 inches. Let's also convert the height to inches.
h= 6 ft * 12 in/1 ft= 72 in
Now, we calculate the total volume of the tank.
To find the volume of 75 % of the tank we multiply the total volume by 0.75.
\begin{gathered}
V_\text{tank}=0.75\cdot V_\text{total}\approx {\color{#FF00FF}{1\,374\,133}} \text{ in}^3
\end{gathered}
We know from Part A that 1206 cubic inches of water are added to the tank every second. Let's convert this to minutes.
\begin{aligned}
V_\text{flow} & ={\color{#A800DD}{1206}} \cancel{\text{ in}^3/\text{s}} \cdot \dfrac{60 \text{ in}^3/\text{min}}{1 \cancel{\text{ in}^3/\text{s}}} \\
& ={\color{#A800DD}{72\,360}} \text{ in}^3/\text{min}
\end{aligned}
Now, we can calculate the time it would take to fill 75 % of the tank.
