Part A.
30 miles east in half hour next
turn left and drive 30 miles north in 1 hour
The average speed is given by
the total distance divided by the total amount of time.
In this case,
[tex]\begin{gathered} \text{average velocity=}\frac{30+30}{1.5+1} \\ \text{average velocity=}\frac{60}{2.5} \\ \text{average velocity=}24\text{ }\frac{miles}{hour} \end{gathered}[/tex]Now, the position in rectangular coordinates is (30,30)
In polar coordinates, we must find the angle theta and the lenght r.
This can be given as
[tex]\begin{gathered} r=\sqrt[]{x^2+y^2} \\ \end{gathered}[/tex]In our case x=30 and y=30, hence
[tex]\begin{gathered} r=\sqrt[]{30^2+30^2} \\ r=\sqrt[]{2\cdot30^2} \\ r=30\sqrt[]{2} \end{gathered}[/tex]On the other hand, angle theta is given by,
[tex]\theta=tan^{-1}\frac{y}{x}[/tex]hence,
[tex]\begin{gathered} \theta=tan^{-1}\frac{30}{30} \\ \theta=tan^{-1}1 \\ hence, \\ \theta=45\text{ degr}ees \end{gathered}[/tex]Therefore, in polar coordinates, point (30,30) is given by
[tex](r,\theta)=(30\sqrt[]{2},45)[/tex]Part B
Until now, we drove 30 miles east, next 30 miles north. Now, we must drive 30 miles west and 30 miles south.
Hence, we will drive 30+30+30+30=4*30=120 miles. Therefore, the average velocity is
[tex]\begin{gathered} \text{average velocity=}\frac{120}{1.5+1+1.5+1.5} \\ \text{average velocity=}\frac{120}{5.5} \\ \text{average velocity=}21.81\text{ }\frac{miles}{hour} \end{gathered}[/tex]Part A. Velocity.
Velocity is a vector. In this case, we must add to vectors:
vector V1 is given by
[tex]\begin{gathered} v_1=(\frac{30}{1.5},0) \\ \text{which is equal to } \\ v_1=(20,0) \end{gathered}[/tex]Vector V2 is given by
[tex]\begin{gathered} v_2=(0,\frac{30}{1}) \\ v_2=(0,30) \end{gathered}[/tex]then, resultant vector velocity is
[tex]\begin{gathered} v=v_1+v_2 \\ v=(20,0)+(0,30) \\ v=(20,30) \end{gathered}[/tex]
