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Related Rates

Related Rates - 2D Geometry


As shown in the above diagram, a sailboat is pulled toward a rock face 30m30\text{m} high by a rope 66m66 \text{m} long. If the rope is pulled at a rate of 8m8 \text{m} per second, what is the speed of the sailboat after 22 seconds?

Suppose a 20 cm×20 cm20 \text{ cm} \times 20 \text{ cm} rectangle is modified such that the width of a rectangle increases at a rate of 1010 cm per second, while the length of the rectangle decreases at a rate of 33 cm per second. What is the ratio Width:Length \text{Width} : \text{Length} when the rate of change of the area of the rectangle is 0?0?

A plane is flying in a straight line parallel to the ground at a constant speed of 400 m 400 \text{ m} per second at an altitude of 4000 m, 4000\text{ m}, directly above an observer on the ground. After 33 seconds, as shown in the above diagram, the observer now looks at the plane at an angle of θ \theta from vertical. If the rate of change of θ \theta with respect to time measured in seconds at this instant is dθdt=ab,\displaystyle{\frac{d\theta}{dt}=\frac{a}{b}}, where aa and bb are positive, coprime integers, what is a+b? a+b?

The radius of a circle starts at 10 cm10\text{ cm} and increases at the rate of 1 mm1\text{ mm} per second. If S(t)S(t) is the area of the circle (in cm2\text{cm}^2) after tt seconds and S(14)=aπ cm2/s,S'(14) = a\pi \text{ cm}^2\text{/s}, what is a?a?

Consider a circular sector with radius rr, central angle θ,\theta, and arc length ll. If the circular sector begins to expand such that drdt=8,dθdt=7, \frac{dr}{dt} =8, \frac{d\theta}{dt}=7, what is dldt\displaystyle{ \frac{dl}{dt}} for θ=π2\displaystyle{ \theta = \frac{\pi}{2}} and r=15? r = 15?


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