# Moms make the best lifeguards

A family decides to go to the lake on their summer vacation. Their child is small, just around 4 or 5 years old and is very excited. When they get to the lake, the child runs immediately into the water while his parents set up their blanket on a little beach by the lake. Out in the lake, the child steps on a rock, hurts his toe, and begins to cry for his mom to come get him.

Mom's velocity is $$v_s=3~m/s$$ on the sand and $$v_w=1~m/s$$ in the water. The distance between the mom and the child is given in the figure, with $$d=30~m$$, $$a=10~m$$ and $$b=30~m$$. What's the minimum time in seconds for her to reach her child?

Details and assumptions

• Remember, the quickest path between 2 points on a plane is a straight line (if you travel at constant velocity).
• There are a couple ways to do this problem, but one straightforward way involves solving a quartic equation, which you can do here.
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