Application Problems
Number One

San Francisco Bay is an inlet of the Pacific Ocean. At a dock, the depth of the water is 3ft at low tide at 2 in the morning and high tide is 71ft, which occurs every 5 hours. Draw a graph showing the depth of the water and write a function.

This problem may look different from the rest, but it is just like the others, except it is in a word problem. In an application problem, you look at the numbers that they give you. In this specific application problem we give you the depth of the water at low tide and what time it is at and the depth at high tide and how long it takes from high tide to high tide agian. First thing we do is realize that we can not go lower than 3 since that is the lowest the water can go. Next thing we see is that the highest we can go is 71 feet. Now we know our low and high points, we need to find the middle line, where our middle points will go. We subtract 3 from 71 and get 68. If we divide that by two, we get 34. So now we will add 34 to 3 and subtract 34 from 71 to get the middle line, or the sinusodial axis, which is at 37. Next we know that a period is 5 hours, so the two low points are at (2,3) and (7,3). Next we have to find the high point. We know it will be at 71, but we need to find the x-axis coordinate. We divide 5 by 2 and get 2.5. Because of that we will add 2.5 to 2, so that means that the high point will be at the coordinates (4.5,71). Next we need to find the mid-points. So now we divide 2.5 by 2 to get 1.25. When we add that to 2 and 4.5, we get 3.25 and 5.75, and we know that it will be on the sinusodial axis. If we subtract 1.25 from 2, we .75, so that will also be on the sinusodial axis as well. Now you can draw the lines and get the graph, but we still need to find the equations. Well since we know the amplitude (34), the period (5), and the sinusodial axis (37). Now we need to see if there is a transformation. Out of any graph, you can make a cosine and a sine equation. If we look to see where the sine graph starts we see that it was moved to the right .75. If you look for a cosine graph, you see that it is moved to the right by 2. Now we can write the equation. Amplitude is first, then sine or cosiine, then B, which you get by doing 2pi/5 (period), which can not reduce, then it is the parenthesis with the transformation, and then last but not least is the sinusodial axis. The amplitude, B, and the sinusodial axis is the same for both a cosine and sine equation. The only difference is the transformation. Now that we know that, lets write the equations. The sin equation is y=-34sin(2pi/5)(x-.75)+37 and the cos equation is y=-34cos(2pi/5)(x-2)+37. If you got those and the graph, then you did what the problem asked you to do.

Number Two

You go to the carnival and decide to ride the Ferris Wheel. The wheel is 3ft off of the ground and the diameter of it is 38ft. It takes you 4 seconds to reach the top. The wheel makes a revolution every 10 seconds. Draw a graph and write a funtion.

Just like the above, it is an application problem. In the problem, they give us the information that we need. It says that the ferris wheel is already off the ground 3 feet when you get on, so we know that none of our points will go lower than 3. The next thing that it tells us is that the diameter of the ferris wheel is 38 feet. That means we will add 38 to 3 to get our high point, which will end up being 41 feet. So now that we know where the high points will hit and where the low point will hit, we have to find in between 41 and 3, or as we learned it, the sinusodial axis. Since it is 38 feet between the two, we have to divide that by 2 and we get 19, which will end up being your amplitude. We will now add 19 to 3 and subtract 19 from 41, which will give us 22. 22 is now our sinusodial axis and where our middle points will be at. The next thing that we look at is that the problem says that it takes you 4 seconds to reach the top, so now know the point (4,41). The last thing that the problem tells us is that it takes 10 seconds to reach the top again. Now we know that it will be at the top, which is 41 feet, 10 seconds after 4 seconds. The next high point is at (14,41). To find the low point of the ferris wheel, we have to divide the distance from the first high point from the second high point by 2. Since it took 10 seconds, the low point will be 5 seconds after the first high point and 5 seconds before the second high point, but it has to be at 3 feet since that is the lowest that you can go on the ferris wheel. Now that we know that, we know that our point is at (9,3). If you plotted those points, you can see that we now have the high and low points, but now we need the points that are between the high points and the one low point. To find that you take 9-4 and you get 5. There is 5 seconds between the points, so now you have to divide that also by 2 in order to find the mid points. Since 5 divided by 2 is 2.5, we will add 4+2.5 to get 6.5 and 9+2.5 to get 11.5 . Since we found where the mid points would go (sinusodial axis), we will now plot them. The points are (6.5, 22) and (11.5,22). Now that we have all of the points plotted, you can draw the lines and get your graph. The problem also said to write an equation. Well since we know the amplitude (19), the period (10), and the sinusodial axis (22). Now we need to see if there is a transformation. Out of any graph, you can make a cosine and a sine equation. If we look to see where the sine graph starts we see that it was moved to the right 1.5. If you look for a cosine graph, you see that it is moved to the right by 4. Now we can write the equation. Amplitude is first, then sine or cosine, then B, which you get by doing 2pi/10 (period), which reduces to pi/5, then it is the parenthesis with the transformation, and then last but not least is the sinusodial axis. The amplitude, B, and the sinusodial axis is the same for both a cosine and sine equation. The only difference is the transformation. Now that we know that, lets write the equations. The sine equation is y=19sin(pi/5)(x-1.5)+22 and the cosine equation is y=19cos(pi/5)(x-4)+22. If you got those and the graph, then you did what the problem asked you to do.

You Try:

A tsunami approaches the San Francisco Bay. The water first goes down from its normal level, then rises and equal distance above its normal level, and finally returns to its normal level. The tsunami that is approaching has an amplitude of 12 meters. The period is about 15 minutes. The normal depth of water at San Francisco Bay is 12 meters. Draw a graph and write a function.