### Quiz 3.10:

Question:

A dinghy is pulled toward a dock by a rope from the bow through a ring on the dock 6 ft. above the bow. The rope is being hauled in at a rate of 2 ft/sec. How fast is the boat approaching the dock when 10 ft of rope are out?

Solution:

Let $$a$$ be the distance from the dock and $$c$$ be the amount of rope out. Both measured in feet.

By the Pythagorean Theorem, we have $$a^2+36=c^2$$

Differentiating that equation implicitly with respect to $$t$$ gives $$2a \frac{da}{dt}=2c \frac{dc}{dt}$$

We were given that $$\frac{dc}{dt}=-2$$ ft/s

We were asked for $$\frac{da}{dt}$$ when $$c=10$$, which by Pythagoras would mean $$a=8$$

So putting that all together get get,

$$2(8)\frac{da}{dt}=2(10)(-2)$$

$$\frac{da}{dt}=\frac{-10}{4}=-\frac{5}{2}$$ ft/s