Question

A ferry transports tourists between three islands. It sails from the first island to the second island,$4.76\mathrm{km}$away, in a direction$37.0°$north of east. It then sails from the second island to the third island in a direction$69.0°$west of north. Finally it returns to the first island, sailing in a direction$28.0°$east of south. Calculate the distance between

(a) the second and third islands and (b) the first and third islands.

Short answer

(a)The distance between 2 and 3 islands is$A=7.17\text{km}$

(b)The distance between 1 and 3 islands is$B=6.15km$

Step-by-step solution

Definition of displacement vector and given equations.

• The vector distance between the starting and end points can also be used to determine an object's displacement.
• It is given that
• Distance between island 2 and island $3\overrightarrow{A}=A$at $159°$
• Distance between island 3 and island $1\overrightarrow{B}=B$at $298°$
• Distance between island 1 and island $2\overrightarrow{C}=4.76\text{km}$at $37°$

Determine the distance between the second and third islands.

(a)

Take the$X-$ axis eastward to get the$X-$components

$\left(4.76\text{km}\right)cos{37}^{°}+Acos{159}^{°}+Bcos{298}^{°}=0$

$(3.80\text{km})-0.934A+0.469B=0$

Then $B=-8.10\text{km}+1.99A$

For thedata-custom-editor="chemistry" $y-$components

$$\begin{gathered} \left(4.76\text{km}\right)sin{37}^{°}+Asin{159}^{°}+Bsin{298}^{°}=0 \\ \left(2.86\text{km}\right)+0.358A-0.883B=0\hspace{0.33em}\hspace{0.33em}\left(1\right) \end{gathered}$$

Now, we have to calculate the distance between island 2 and island 3 , A

By substituting the value of B into equation (1), we get

$$\begin{gathered} (2.86\text{km})+0.358A-0.883(-8.10\text{km}+1.99A)=0n \\ 2.86\text{km}+0.358A+7.15\text{km}-1.76A=0n10.0\text{km}=1.4 \\ \text{Then}A=7.17\text{km} \end{gathered}$$

Therefore, the distance between 2 and 3 islands is $A=7.17\text{km}$

Determine the distance between the first and third islands.

(b)

Now, we have to calculate the distance between island 3 and island 1 , i.e., B

Then

$$\begin{gathered} B=(-8.10\text{km})+1.99A \\ =-8.10\text{km}+(1.99)(7.17\text{km}) \\ =6.15\text{km} \end{gathered}$$

Therefore, the distance between 1 and 3 islands is$B=6.15km$