Question

The position of an open-water swimmer is shown in the graph. The shortest route to the shoreline is one that Ay 10 8 water 6 4 (3, 2) • swimmer 2 1 1 2 4 5 x shore 2 An equation that represents the shortest path is y =

Answer 1

The shortest route is the one that is perpendicular to the shore.

In xy-coordinates, this means that the slope of the shore line and the slope of the shortest route line are negative reciprocals.

The shortest route line has also to pass throught the point where the swimmer is, that is (3,2).

We start finding the slope of the shore line. To find it, we pick two point of the line, like (0,5) and (2.5,0) and calculate:

`m=(y_2-y_1)/(x_2-x_1)=(5-0)/(0-2.5)=(5)/(-2.5)=-2`

The negative reciprocal of this slope is:

`m_p=(1)/(-m)=(1)/(-(-2))=(1)/(2)`

Then, we know that the shortest route line has a slope of mp=1/2.

With point (3,2) we can write the slope-point form of the line and rearrange into a slope-intercept form:

`\begin{gathered} y-2=(1)/(2)(x-3) \\ y-2=(1)/(2)x-(3)/(2) \\ y=(1)/(2)x-(3)/(2)+2 \\ y=(1)/(2)x+(1)/(2) \end{gathered}`

Answer: The shortest route can be represented by y=1/2*x+1/2