Do you have an answer? A few days ago a challenge problem was offered here. It involved a lifeguard at a beach who wanted to reach a struggling swimmer in the quickest amount of time. What value of X would allow the least time? Running speed in the sand was 4.5 m/s. Swimming speed was 2.4 m/s.
Two Basic Concepts Needed
First, you need to use the Pythagorean Theorem where the hypotenuse squared equals the sum of the squares of the other two sides. The guard needs to follow two distances shown in red in order to reach the swimmer. The first distance d1 is the hypotenuse of the upper left triangle in the sand. The second distance d2 is the hypotenuse of the lower right triangle in the water. In the two triangles, we would write (d1)2 = 302 + X2 and (d2)2 = 402 + (50-X)2. We don’t know what value of X solves the problem. But, we can still express the concepts in terms that include the unknown X.
The second concept needed is about distance, rate, and time. You know how that works. If you drive a distance of 200 km at a rate of 50 km/hr, it will take a time of 4 hr to reach your destination. You divide distance by the rate to get the time. We will use that idea here to find the total time T for the two parts of the problem. In the statements below, values of v1 and v2 are the running and swimming rates of the guard. The last line of the expression needs the 4.5 m/s and 2.4 m/s rates to make it complete.
Three Ways To Get The Value Of X
First, there is brute force of calculation. Put in the values for the running and swimming rates in the last expression above. Try a value of X by guessing. Write it down. Try another one, and another, etc. In time, you will begin to narrow down the result by noticing answers that are smaller than others. It is a slow and inefficient method.
Second, use the computing power of a spreadsheet to do those many trial calculations for you. It is quicker and can yield a value for X that is pretty close to correct. Yes, you need to know how to work with spreadsheets in order to use this method. That isn’t trivial. I’ve included a screen shot of the first part of the spreadsheet from my computer.
The highlighted cell under the T was calculated using the function fx in the entry bar that starts with =((900 +. The formula for fx was copied down the T column into each cell. The values in the column X are test values for the guard’s entry into the water. The spreadsheet includes values for X to 50.
Doing it this way speeds up the calculations. Values for X can be adjusted and changed as one narrows in on the minimum total time T in the next column. You can even try different running and swimming speeds if you want. Spreadsheets will also allow charts to be made to visualize the results. It is a powerful tool.
Recently, I noticed a chart in an article that used data from a spreadsheet. The chart allowed the reader to interact with it and explore values on it. It was called plotly. The online site allows the user to upload a spreadsheet and display the data in a wide variety of charts. I created this one using plotly from my spreadsheet mentioned above. Click on the blue chart to be taken to the url. You can then move your mouse around over it to answer the question for the least time for the rescue by the lifeguard. It will give you a reasonable, but not exact, answer.
Third, this method utilizes a great graphing application on my Apple computer called Grapher. With it, I entered the equation generated in the earlier part of this post for total time T. It looked a lot like what you see within the big brackets (….) in the image below. Grapher generated a plot instantly just like the blue one above. The challenge question is asking for the value of X that makes the graph above reach exactly its lowest value. At that point, it would have a slope = 0.
In calculus, the first derivative of a function is the slope of the function. The application Grapher is able to calculate the derivative and plot the resulting function. That has been done below. The screen shot shows a line crossing the x axis with a value = 0 where a dotted line runs vertically. The bottom of the image shows x = 32.85957 meters as the solution to the lifeguard problem.
Thanks for joining me on this challenge. I hope you didn’t get a bad sunburn.