# Math 206

**Topics:**English-language films, Arc length, Southern United States

**Pages:**3 (763 words)

**Published:**October 30, 2012

Dear Professor Anderson, CEO of CompuCalc,

Lord Greene wants us to help his grandson, 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,...

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