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\problemgrouptitle{Related Rates}

\begin{problemblock}{Solve the following problems involving related rates.}

\item An oil spill spreads in a circle whose area is increasing at a constant rate of 10 square kilometres per hour. How fast is the radius of the spill increasing when the area is 18 square kilometres?%
\answer{$\displaystyle\frac{5}{\sqrt{18\pi}}$ km/h}

\item A spherical balloon is being filled with water at a constant rate of 3~cm$^3$/s\,.  How fast is the diameter of the balloon changing when it is 5~cm in diameter?
\answer{$\displaystyle \frac{6}{25\pi}$ cm/sec}

\item An observer who is 3~km from a launchpad watches a rocket that is rising vertically.  At a certain point in time the observer measures the angle between the ground and her line of sight of the rocket to be $\pi/3$ radians.  If at that moment the angle is increasing at a rate of $1/8$ radians per second, how fast is the rocket rising when she made the measurement?
\answer{$\displaystyle\frac{3}{2}$ km/sec}

\item A water reservoir in the shape of a cone has height 20 metres and radius 6 metres at the top. Water flows into the tank at a rate of 15~m$^3/\min ,$ how fast is the level of the water increasing when the water is 10~m deep? Hint: Use similar triangles.\textbook{ (See Section~\ref{subsec:functiontypes})}
\answer{$\displaystyle \frac{5}{3\pi}$ m/min}

\item At 8 a.m., a car is 50 km west of a truck. The car is traveling south at 50~km/h and the truck is traveling east at 40 km/h. How fast is the distance between the car and the truck changing at noon?
\answer{$\displaystyle \frac{1840}{29}$ km/h}

\item A boy is walking away from a 15-metres high building at a rate of 1~m/sec . When the boy is 20~metres from the building, what is the rate of change of his distance from the top of the building?\answer{$\displaystyle \frac{4}{5}$ m/sec}

\item The hypotenuse of a right angle triangle has a constant length of 13~cm. The vertical leg of the triangle increases at the rate of 3~cm/sec. What is the rate of change of the horizontal leg, when the vertical leg is 5~cm long?
\answer{$\displaystyle -\frac{5}{4}$ cm/sec}

\end{problemblock}
