Question

The position of an open-water swimmer is shown in the graph. The shortest route to the shoreline Ay 10 8. 6. water 4 shore (2, 1) • swimmer 19 2 1 12 3 4 5x -2 An equation that represents the shortest path is y=0.

Answer 1

First, we have to find the slope of the given line.

`m=(y_2-y_1)/(x_2-x_1)`

Now, we select two points on the line: (0,3) and (1,7).

Then, we replace these points in the slope formula above.

`m=(7-3)/(1-0)=(4)/(1)=4`

If the shorter path is perpendicular, then its slope can be found using the rule for perpendicularity:

`\begin{gathered} m\cdot m_1=-1 \\ m\cdot4=-1 \\ m=-(1)/(4) \end{gathered}`

Assuming that the shorter path passes through the point (2,1), we use the point-slope formula to find the equation:

`y-y_1=m(x-x_1)_{}_{}`

But, we have to replace the slope m = -1/4, and the point (2,1) in the formula above:

`y-1=-(1)/(4)(x-2)`

Now, we use the distributive property to get rid of the parenthesis.

`y-1=-(1)/(4)x-2(-(1)/(4))`

We solve the product.

`y-1=-(1)/(4)x+(2)/(4)`

Then, we simplify the fraction 2/4.

`y-1=-(1)/(4)x+(1)/(2)`

Now, we add 1 on each side.

`\begin{gathered} y-1+1=-(1)/(4)x+(1)/(2)+1 \\ y+0=-(1)/(4)x+(1)/(2)+1 \end{gathered}`

We use the least common factor 2 to solve the sum of the independent terms:

`\begin{gathered} y=-(1)/(4)x+(1\cdot1+1\cdot2)/(2) \\ y=-(1)/(4)x+(1+2)/(2) \\ y=-(1)/(4)x+(3)/(2) \end{gathered}`

Finally, as you can observe, the equation that represents the shortest route, perpendicular to the given line, is

`y=-(1)/(4)x+(3)/(2)`

This line has a slope of -1/4, and its y-intercept is at 3/2, or (0, 1.5).