A sailboat is sitting at rest near its dock. A rope attached to the bow of the boat is drawn in over a pulley that stands on a post on the end of the dock that is 5 feet higher than the bow. If the rope is being pulled in at a rate of 2 feet per second, how fast is the boat approaching the dock when the length of rope from bow to pulley is 13 feet?

To start: use pathagoren therom from the picture.
x²+y²=5²
x²+13²=5²
x²=5²+13²
x=√194

Then differentiate both sides: x²+y²=5²
2x*dx/dt=2y*dy/dt
=(2√194(2))/2(13)=dy/dt
dy/dt=2.14

The boat is approaching the dock at a rate of 2.14 ft/sec.

Problem 3. Point $A$ is 30 miles west of point $B$. At noon a car starts driving South from point $A$ at a rate of 50 mi/h and a car starts driving South from point $B$ at a rate of 70 mi/h. At 2:00 how quickly is the distance between the cars changing?

From the picture:
x= distance travele by car 1
y=distance traveled by car 2
dx/dt=50mi/h
dy/dt=70mi/h
z=distance between the two cars

Differentiate:
z²=(y-x)²+30²
2z*dz/dt=2(y-x)²+30²
2z*dz/dt=2(y-x)(dy/dt-dx/dt)
z*dz/dt=(y-x)(dy/dt-dx/dt)

At 2pm the time is 2 hours
x=50*2=100
y=70*2=140
z²=(y-x)²+30²
z²=(140-100)²+30²
z²=40²+30²
z²=1600+900=2500
√2500=50

Then you must plug in the 50=z
(50)*dz/dt=(140-100)(70-50)
dy/dt=(40*70)/50=16 mi/h
The distance between the cars is changing at 16 mi/h