Three Act Play, Third Act

Here is the student version of the third act

The third act of the three act play both verifies the solution for the original problem as well as extends the problem for different solutions. In verifying our solution, we first had to consider all of the information given to us.

• The sink is 16″ by 14″
• The sink is 7″ deep
• The diameter of the stopper is 3.5″
• The depth of the stopper is 1.5″
• It takes 7.16 seconds to fill a 20 oz bottle

To start in verifying our solution, let us look at the volume of our sink. Since we are given the dimensions of the sink, finding the volume is fairly simple. By multiplying the length and width of the sink by the depth, we find that the sink is . Due to the fact that we are dealing with volume, we will be using cubic inches as our unit of measure. Next, we need to find the volume of the stopper. Because we are given the diameter of the stopper, we can find the area of the circle it forms. Recall that the formula for finding the area of a circle is . Using this formula we see that the area of the circle formed by the stopper is 9.62″. Then, in order to find the volume of the stopper, we multiply the area of the circle by the depth of the stopper, 1.5″. As a result, the area of the stopper is . By adding the volume of the stopper to the area of the sink, we can then find the total area of the sink, which ends up being .

Now that we have the total volume of the sink, let’s look at the rate at which the sink is being filled with water. Using the rate at which the sink filled the 20 oz bottle, we can determine how many cubic inches of water is being poured every second. To do this, we need to find the conversion rate between fluid ounces and cubic inches of water. By doing a quick search using Wolfram Alpha, we see that 1 fluid ounce is of water. Next, we multiply the 20 oz by the conversion factor of 1.805 to convert the ounces into cubic inches, which ends up being . Now that we have converted the flow rate of water from fluid ounces to cubic inches, we can divide the new figure by 7.16 seconds (the time it took to fill the 20 oz bottle), in order to see how many cubic inches of water flows out of the sink in 1 second. After doing some division, it turns out that for every 1 second, of water flows out of the sink.

Finally, now that we have the total volume of the sink and the flow rate of the sink in the proper unit of measure, we can find exactly how long it is expected to take in order to fill the entire sink. To find this, we divide the total volume of the sink ( ) by the rate of water coming out of the faucet every second ( ). This results in us expecting the sink to be filled in 313.912 seconds. By dividing this figure by 60, we see that it will take 5.232 minutes. We can also change the decimal from the 5.232 minutes to seconds by multiplying .232 by 60, which results in the final expected time to fill the sink being 5 minutes and 13.92 seconds.

Comparing this result to the time found in the picture, we see that there is a 29.69 second difference. While this is slightly off of our actual time, first consider the different factors that may have contributed to our predicted time being different from our observed result.

• The rate of water flowing out of the faucet is not always constant
• The measurements we took did not considered the curved corners of the sink and the sloping bottom. Our measurements assumed that the sides were perfectly straight.
• The stopper being plugged is not a perfect countermeasure for preventing water from draining out as we fill the sink.

Given the differences between our simulation and the reality we observed in the video, it should be no surprise that our result should be different from the one we observed. However, from our simulation we did find that a sink of that approximate size should take around 5 minutes to fill.

An extension or sequel for this type of problem could involve finding the area of the curved curved corners of the sink in order to have even more precise measurements for the volume. Another option would be finding how long it take to fill a sink with different dimensions or different rate of water coming from the faucet.