Question 6 (a) Near a surface buoy the depth of a lake at the point with coordinates (x, y) is given by z = 100+ 0.05z² -0.02y² where x, y and z are measured in metres. The coxswain driving the small boat starts at the point (60, 30) and moves towards the buoy, which is located at (0,0). Is the water under the boat getting deeper or shallower when he departs? At what rate? Explain your answer. (b) Determine the equation of the tangent plane to the surface z+1=re" cos z at the point (1,0,0).

Findl DF of fex,y)=x²e° at the point (-2,0)-

Consider a denolition crane with a 35-pound ball suspended from a 40-foot cable that weighs 6 pounds per foot. Exercise (a) Find the work required to wind up 15 feet of the apparatus. Step 1 Note that in the current situation, as the cable is pulled, the following things happen. 1. The ball gets lifted with a constant constant force. 2. The cable that is wound up gets lifted with a variable variable force. 3. The cable that doesn't get wound up is lifted with a constant constant force. Step 2 Calculate the work done, W,, in pulling up the ball while winding up 15 ft of the apparatus. W, = force x distance = 35 x 15 = 525 525 ft-lb Step 3 Consider an elemental length Ay, of the cable that gets wound up. As the cable weighs 6 pounds per feet: Weight = 6 Ay. Suppose it is at a vertical distance y from the winding engine, then Distance moved = (6 15 -) ft. a e Type here to search