How do i approach this word problem?
Say that you are in a pirate ship that is traveling along a curved river (which roughly follows the equation $y_1=x_1(sin(x_1)+1)$) as you travel from the south-west and you head to the north-east (from $-\infty<x_1<\infty$). The cannons out of the side of your ship can only fire perpendicular to the direction that the ship is pointing. Ahead, you see a castle tower (the outer walls of the castle are given by the parametric equation $(x_2,y_2)=(cos(t)-1,sin(t)+3)$ or the implicit equation $(y_2-3)^2+(x_2+1)^2=1$). As you float down the river you know that you might only get a few chances to fire on the castle and you want to do that from the closest possible points. The object of this problem is to find those points where you should fire on the tower.
Determine the points as you pass along the river where you will get the best shot on the castle tower. Do this by minimizing the equation of the distance between a point on the river and a point on the castle tower subject to the constraint that the slope of the line between the points must be perpendicular to the river. Make clear what function you are trying to minimize and what your constraint equation is.
Also, determine the points along the curve following the river and the points on the castle tower that you will hit. Determine if your points are maxima or minima and pick out the ones that minimize the distance.