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# Related Rates

How does pressure change in a combustion engine? How fast does the water level rise when filling a pool? Calculus quantifies the impact of change on areas, angles, distances, temperatures, and more.

# Related Rates - 2D Geometry

As shown in the above diagram, a sailboat is pulled toward a rock face $$30\text{m}$$ 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?

Suppose a $$20 \text{ cm} \times 20 \text{ cm}$$ rectangle is modified such that the width of a rectangle increases at a rate of $$10$$ cm per second, while the length of the rectangle decreases at a rate of $$3$$ cm per second. What is the ratio $\text{Width} : \text{Length}$ when the rate of change of the area of the rectangle is $$0?$$

A plane is flying in a straight line parallel to the ground at a constant speed of $$400 \text{ m}$$ 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 $$a+b?$$

The radius of a circle starts at $$10\text{ cm}$$ and increases at the rate of $$1\text{ mm}$$ per second. If $$S(t)$$ is the area of the circle (in $$\text{cm}^2$$) after $$t$$ seconds and $$S'(14) = a\pi \text{ cm}^2\text{/s},$$ what is $$a?$$

Consider a circular sector with radius $$r$$, 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 $$r = 15?$$

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