1) Let's call [tex]V_S[/tex] the speed of the southbound boat, and [tex]V_E=V_s+3~mph[/tex] the speed of the eastbound boat, which is 3 mph faster than the southbound boat. We can write the law of motion for the two boats:
[tex] S_E(t)=V_E t=(V_S+3)t[/tex]
[tex]S_S(t)=V_S t[/tex]
2) After a time [tex]t=3~h[/tex], the two boats are [tex]45~mi[/tex] apart. Using the laws of motion written at step 1, we can write the distance the two boats covered:
[tex]S_E(3~h)=3(V_S+3)=3V_S+9[/tex]
[tex]S_S(3~h)=3V_S[/tex]
The two boats travelled in perpendicular directions. Therefore, we can imagine the distance between them (45 mi) being the hypotenuse of a triangle, of which [tex]S_E[/tex] and [tex]S_S[/tex] are the two sides. Therefore, we can use Pythagorean theorem and write:
[tex]45= \sqrt{(3V_S)^2+(3V_S+9)^2} [/tex]
Solving this, we find two solutions. Discarding the negative solution, we have [tex]V_S=9~mph[/tex], which is the speed of the southbound boat.
