River Skipper Dear Professor Anderson
Little Johnny, with his new toy, the River Skipper boat. According to Lord Greene, Little Johnny likes to play around the center of the Greene Hilly estate, near the South Bridge. He has to release his boat somewhere along the Raging River and Lord Greene wants to suggest that he put it near the South Bridge and then retrieve the boat at the North Bridge. However, Lord Greene is concerned if Little Johnny will reach the North Bridge in time to retrieve the boat from the river.
Therefore, we need to find out how fast Little Johnny has to travel to retrieve the boat, whether it be in a straight path through the woods or along the driveway. Also, Lord Greene’s engineers informed us about the velocity setting that is currently being used for the River Skipper, V1 (x) = 1+5 cos2 x2, but they are also considering to use V2 (x) = 5 cos2 x2. Our primary objective is to find the fastest rate for Little Johnny to travel to the North Bridge and to also find his rate if he were to travel along the driveway. To find his rate, we will use the formula distance = rate x time.
However, the difficulty of this problem is finding the time that it takes the River Skipper to get from the South Bridge to the North Bridge with each velocity setting. Finding the time is difficult because the rate of the River Skipper is always changing at each position. To overcome this difficulty, we found the area under the curve of the Raging River and subdivided it into smaller intervals. This overcomes the difficulty because the rate is considered to be fairly constant within each smaller piece of area and results in an approximation of the rate of the River Skipper.
First, we used the distance between two points formula (x2-x1)2+(y2-y1)2 to find the distance between the South Bridge and the North Bridge to determine Little Johnny’s straight path. Additionally, we used the arc length formula 1+(f? x)2 to determine the distance along the driveway. Following that, we subdivided the Raging River (y = sinx) into 1000 intervals between the South Bridge (0,0) and the North Bridge (0. 877, 0. 769). The distance of each interval was found by using the distance between two points formula and the right endpoint of each interval was used for the position.
The position of each interval was inputted into each of the velocity settings. Then, the distance between two points of each interval was divided by each of the velocity settings using position as a function of x. Using this formula, 1000 intervals were summed together resulting in the time that it took the River Skipper to travel along each interval. Using the calculated time of the River Skipper and the distances of the two different paths, we used r = d/t to find the rate that Little Johnny had to travel.
As a result, the distance between the South Bridge and the North Bridge through the woods, in a straight path, is 1. 7 and along the driveway is 1. 22. The total time that it took the River Skipper to travel by using the first velocity setting is 0. 216 and for the second velocity setting is 0. 267. Therefore, using the first velocity setting, Little Johnny would run in a straight path at a rate of 5. 42 and along the driveway, he would run at a rate of 5. 65. Using the second velocity setting, he would run in a straight path at a rate of 4. 39 and along the driveway at a rate of 4. 57.
Distance between South Bridge and North Bridge * Straight Path – 1. 17 * Along Driveway – 1. 22 * Total travelling time of River Skipper V1 – 0. 216 * V2 – 0. 267 * Rate of Little Johnny using V1 * Straight Path – 5. 42 * Along Driveway – 5. 65 * Rate of Little Johnny using V2 * Straight Path – 4. 39 * Along Driveway – 4. 57 Overall, we feel that Little Johnny would run faster in a straight path through the woods at a rate of 4. 39 with the River Skipper set at the second velocity setting. In doing so, Little Johnny will be able to reach the North Bridge before his boat arrives. Also, if he were to travel along the driveway, he would have to run faster to retrieve his boat since his rate would be.