Hello, FalconPUNCH!

You left out a measurement in #2.

. . I'll pick a convenient value . . .

2. A boat is pulled into a dock by a rope attached to the bow of the boat and

passing through a pulley on the dock that is **6** m higher than the bow of the boat.

If the rope is pulled in at a rate of 1 m/s,

how fast is the boat approaching the dock when it is 8 m from the dock? Code:

* P
* |
R * |
* | 6
* |
* |
B * * * * * * *
x

The boat is at $\displaystyle B$, the pulley is at $\displaystyle P.$

The length of the rope is: .$\displaystyle R \,=\, BP$ . and . $\displaystyle \frac{dR}{dt} \,=\, -1\text{ m/s}$

From Pythagorus, we have: .$\displaystyle x^2 + 6^2 \:=\:R^2$

Differerentiate with respect to time: .$\displaystyle 2x\left(\frac{dx}{dt}\right) \:=\:2R\left(\frac{dR}{dt}\right)$

. . and we have: .$\displaystyle \frac{dx}{dt} \:=\:\frac{R}{x}\left(\frac{dR}{dt}\right)$

Can you finish it now?