Chapter 4.8, Problem 29PPS

(a)

To determine

To find: The equations of the line along which first and second vehicles lie.

Answer to Problem 29PPS

The second vehicle travel along line $y=-x$ .

Explanation of Solution

Given information: The first and second vehicles stop about $300yards$ from the dock while third one stops about $212yards$ from the dock.

  

The second craft traveled from the origin in a directly northwest direction.

Hence, it would be analogous to traveling along the line $y=-x$ .

  

(b)

To determine

To prove: The first and second vehicle formed an isosceles triangle.

Explanation of Solution

Given information: The first and second vehicles stop about $300yards$ from the dock while third one stops about $212yards$ from the dock.

  

Proof: The boats travel equal path north and east/west.

Hence, with substitution $N=EorN=W$

  $\begin{array}{l}D=\sqrt{N^2+N^2}\\ D=\sqrt{2}.N\end{array}$

Here, $\sqrt{2}.N$ is the total distance from the origin that both boats travelled.

Therefore, distance between both of them,

  $\begin{array}{l}D=\sqrt{{\left(\sqrt{2}.N\right)}^2+{\left(\sqrt{2}.N\right)}^2}\\ =2N\end{array}$

Here, we can see that $E+W$ are the distance between both of them.

Therefore, by symmetry, it is a right isosceles triangle.

(c)

To determine

To find: The coordinates of the locations of all three watercrafts.

Answer to Problem 29PPS

The coordinates of the locations of all three watercrafts are $\left(\frac{300}{\sqrt{2}},\frac{300}{\sqrt{2}}\right),\left(\frac{-300}{\sqrt{2}},\frac{300}{\sqrt{2}}\right)$ and $\left(0,212ft\right)$ .

Explanation of Solution

Given information: The first and second vehicles stop about $300yards$ from the dock while third one stops about $212yards$ from the dock.

  

Calculation: The third watercraft went directly north,

Hence, its location is $\left(0,212ft\right)$ .

The other two watercrafts went a total distance of $300ft$ ,

  $\begin{array}{l}\therefore 300=\sqrt{2}.N\\ \therefore N=\frac{300}{\sqrt{2}}\end{array}$

Therefore,

The coordinates of the location for the first watercraft $=\left(\frac{300}{\sqrt{2}},\frac{300}{\sqrt{2}}\right)$

The coordinates of the location for the second watercraft $=\left(\frac{-300}{\sqrt{2}},\frac{300}{\sqrt{2}}\right)$

(d)

To determine

To prove: The position of all three watercraft is approximately collinear and third craft is at the midpoint of other two.

Explanation of Solution

Given information: The first and second vehicles stop about $300yards$ from the dock while third one stops about $212yards$ from the dock.

  

Proof: The third watercraft went directly north,

Hence, its location is $\left(0,212ft\right)$ .

The other two watercrafts went a total distance of $300ft$ ,

  $\begin{array}{l}\therefore 300=\sqrt{2}.N\\ \therefore N=\frac{300}{\sqrt{2}}\\ \therefore N\approx 212ft\end{array}$

Therefore, all the boats are extremely close to the line $N=212ft$ .