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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.

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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