Filling the Pool

by Math5

As a culminating project for our geometry unit, my students engaged in the following project. This project required that they organize their problem-solving approach, collaborate with each other, and apply many of the skills and concepts developed in 5th Grade Mathematics this year. The following is the account of this work written by the students themselves.

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One day, we walked into math class and we saw a question posted on the board:

“How long does it take to fill the pool?”

At first we thought that it would be easy to find out. After all, we had a lot of practice finding volume of a rectangular prism. We all immediately left for the pool, to get our measurements. As we walked over and as we started taking our measurements, we realized that we couldn’t just take the measurements, because there was a curve at the bottom of the pool. That’s where most of our project came in. Suddenly the simple question on the board became a project that we worked on for a week and a half.  

Estimating the Volume of the Pool

We came up with four different ways to estimate the volume of the pool.

For the first estimate we made, we had to pretend that the pool was broken up into smaller rectangular prisms. So the bottom of the pool looked like a staircase. We multiplied the depth of each prism times the width of the pool times the length of each prism.

The individual prisms were:

1.3m x 14m x 2.6m= 47.32m3

1.4m x 14m x 5m= 98m3

1.5m x 14m x 3.9 = 81.9m3

1.9m x 14m x 3.8m=103.74m3

2m x 14m x 7m= 196m3

1.9m x 14m x 1m=26.6m3

When, we added up all of the volumes, we got the estimated volume, 553.56m3.

For the second estimate, we built a model of the pool out of blocks representing one cubic meter. The first layer of blocks was 23x14 cubes (the surface of water exposed) and the second layer was 18x14 cubes. When we added all the cubes up we got 574 cubes. So our third estimation for the volume is 574 cubic meters.

In the third estimate we used mean depth to figure out the volume. We found the mean depth by adding all the depths of the pool and then dividing it by the amount of depths (6). We then used the length of the pool, the width of the pool, and the mean depth of the pool, 1.7 m to find the volume of a rectangular prism. We multiplied 14•23•1.7 and got 547.4 cubic meters. So our third estimate for the volume is 574 cubic meters.  

For the fourth estimate we turned the whole pool into a trapezoidal prism. Then we flipped it 90° onto its east side. We first needed to find the area of the trapezoidal base. 3.3 meters is the sum of the lengths of two bases of the trapezoidal face of the pool (the parallel sides). These are really the east and west walls of the pool, but to make it easier to think about, we’re thinking of them as the top and bottom base of the trapezoidal prism. The height of the trapezoid is length of the pool (23 m). To find the area: (3.3 • 23) / 2 = 37.95 m2 for “Base area.” Volume is Base Area • Height of Prism: 37.95 • 14 = 531.3 m3.

Later, we figured out the mean of all of our estimated volumes, and found an approximate answer of how many cubic meters are in the pool.

Depending upon the method we used we got 4 different estimates for the volume of the pool

1. Breaking the pool into rectangular prisms based on the depths, we got 553.56m3

2. Rounding the depths, we got 574m3

3. Finding the mean depths, we got 547.4m3

4. Thinking of the pool as a trapezoidal prism, we got 531 .3m3

The 1st estimate + the 2nd estimate + the 3rd estimate + the 4th estimate / 4 (the number of estimates) =  551.565 m3 = mean average of the pool volume.

Measuring Flow Rate

Measuring how much time it takes water to fill a certain volume.

We found the time it took to fill a 1 liter graduated cylinder 5 times and averaged it. We found a mean of 9.868 seconds to fill 1 liter from the classroom sink. The way we found this rate was we had one person holding the cylinder under the running water, one person had a stopwatch, and one person watch the water level and told the timer when to click stop. We took 5 trials and the times we got were: 9.603, 10.606, 9.067, 9.816, and 10.248 seconds. Measuring was hard because the bubbles in the water threw us off. The mean was 9.868 seconds. Since 1000 liters equals one cubic meter we multiplied 9.868 by 1000 and got 9,868 seconds to fill one cubic meter. Since 10,000 seconds doesn’t mean much, we converted this to bigger time units.

9,868 seconds / 3600 = 2.741 hrs

That means it would take 2.741 hours to fill one cubic meter of water using the classroom sink.

Since we found that there were 551.565 cubic meters of water in the pool, we can multiply the volume by time to fill one cubic meter to find out how long it would take to fill the pool.

551.565 m3 • 2.741 hrs/m3 = 1511.840 hrs to fill the pool with the classroom sink.

1511.840 / 24 = 62.993 days or about 2 months!

We all know that you don’t fill a swimming pool with a classroom sink. We thought Facilities probably had a bigger pipe to fill the pool.

We wanted to know what Facilities thought of our investigatigation. We composed an email with all of our information and a question for them that said “We want to know,  How long does it take to fill the pool with the equipment that you guys fill the pool with?” They replied the next day. The email said “We have two water feeds in the pool, one is a 2 inch pipe , and one is a 1 inch.  The one inch fill line is attached to a float, so it goes on automatically when the water is low.”  That was really helpful.

All we had to do now was divide the volume by the volume per minute.  The volume of the pool is 551.565 cubic meters, as we learned earlier, and the volume per minute with the two inch pipe is 127 gallons per minute.  We were using liters, so we had to translate 127 gallons into liters, which is 480.747 liters.  We had to turn 551.565 cubic meters into liters too, which is 551,565 liters.  So, finally, we divided 551,565/480.747 which is 1,147.308 and got our answer.  But again, it is a very confusing way of reading time, so we divided it by 60, which is about 19 hours.  So it would take about 19 hours to completely fill the pool using the pool’s plumbing.

Then, of course, Michael wasn’t satisfied and tossed us another question: “What if the pool level had dropped by 2” and the refill pipe (1” diameter) had to bring the pool back to full level, how long would that take?”

Evan from Facilities informs us that there are two pipes used to fill the pool. The primary pipe is a 2 in diameter and would supply 127 gallons per minute. The second pipe, a 1 inch pipe at the top of the pool is used to automatically refill the pool when the level drops more than 2”.  

The first step was finding the volume of the missing water.  We knew that the width and length are 14 meters and 23 meters, but we needed to convert 2 inches to metric.  That happens to be 0.0508 meters, so we proceeded to do the multiplication problem: 0.0508x14x23 which equals 16.3576 m3.  So we knew the volume of the missing water.  

Next was the time to replace the missing water.  That could be figured out by dividing the amount the volume by the volume per minute.  We transferred 16.3576 cubic meters into liters which is 16357.6 liters.  Then we transferred 37 gallons into liters too, which is 140.06024 liters.  So 140.06024 liters per minute.  Finally we divided 16357.6/140.06024 which is 116.789747040274 and got our answer.  The only thing was, there are easier ways of writing time than 116 minutes, so we wrote 1 hour 56 minutes 48 seconds instead.  

We had a lot of fun doing this project and it was a huge learning experience.  It is a really fun project and hopefully we will get to do things like this in the future because it was a fun experience  to us kids.