express the time T that it takes to go from the island to town as a function of the distance x from P to where the person lands the boat

Question

An island is 6 miles from the nearest point P on a straight shoreline. A town is 12 miles down the shore from P. If a person can row a boat at an average speed of 2 miles per hour and the same person can walk 4 miles per hour, express the time T that it takes to go from the island to town as a function of the distance x from P to where the person lands the boat.

how long will it take to travel from the island to town if the person lands the boat 3 miles from P.

Answer 1

I will try to explain this without a diagram. Let me first try to describe the diagram. This sounds like it's part of a max/min problem.

 

Let's call the island point I, draw a vertical line to point P on the shore represented by a horizontal line. 12 mils to the right( could be to the left and it wouldn't matter) is the town point W. I want to find where to land my boat on the shore that would minimize my time, so let's call Point L where I'm going to land. That point L is between P and W,x miles from point P and 12 - x miles from town W. 

 

We need to convert everything into time measurements so R*T=D or T=D/R.

So IP = 6/2 =3

so PL = x/4

so LW = (12 - x)/4

and IL = sqrt(9 + (x^2)/16 )  Pythagorean theorem on triangle ILP.

 

So I will travel from I to L and then continue on to W. So my function is

 

T(x) = sqrt(9 + (x^2)/16) + (12 - x)/4

 

Hope this helped.

Answer 2

Click on the link below to see the geometry of Ira's solution:

 

http://www.wyzant.com/resources/files/291994/geometry_for_island_problem