Calculus
# Related Rates

$30\text{m}$ high by a rope $66 \text{m}$ long. If the rope is pulled at a rate of $8 \text{m}$ per second, what is the speed of the sailboat after $2$ seconds?

As shown in the above diagram, a sailboat is pulled toward a rock face$400 \text{ m}$ per second at an altitude of $4000\text{ m},$ directly above an observer on the ground. After $3$ seconds, as shown in the above diagram, the observer now looks at the plane at an angle of $\theta$ from vertical. If the rate of change of $\theta$ with respect to time measured in seconds at this instant is $\displaystyle{\frac{d\theta}{dt}=\frac{a}{b}},$ where $a$ and $b$ are positive, coprime integers, what is $a+b?$

A plane is flying in a straight line parallel to the ground at a constant speed of$r$, central angle $\theta,$ and arc length $l$. If the circular sector begins to expand such that $\frac{dr}{dt} =8, \frac{d\theta}{dt}=7,$ what is $\displaystyle{ \frac{dl}{dt}}$ for $\displaystyle{ \theta = \frac{\pi}{2}}$ and $r = 15?$

Consider a circular sector with radius