Lesson 58 — Related Rates

A spherical balloon is inflated at 50 cm³/s. What is the rate of change of the radius when $r = 5$ cm?

Same spherical balloon, $dV/dt = 100$ cm³/s. What is $dr/dt$ when $r = 10$ cm?

The radius of a circular disk is increasing at 0.1 m/s. What is the rate of change of the area when $r = 2$ m?

The edge of a cube is growing at 1 cm/s. What is the rate of change of the volume when the edge is 5 cm long?

The side of a square is growing at 2 cm/s. What is the rate of change of the area when the side is 10 cm long?

A ladder of length $L = 5$ m leans against a wall. Its foot slips outward at $0.5$ m/s. What is the rate of descent of the top when the foot is 3 m from the wall?

A ladder of length $L = 10$ m. Its foot slips outward at 1 m/s. What is the rate of descent of the top when the foot is 6 m from the wall?

An inverted conical tank has a top radius of 3m and a height of 6m. Water enters at 4 m³/min. What is $dh/dt$ when $h = 3$ m?

A cylindrical tank has radius $r = 4$ m. Water enters at 2 m³/h. What is $dh/dt$?

Car A starts north at 60 km/h and Car B starts east at 80 km/h from the same intersection. What is their rate of separation after 30 min?

A lamppost is 4m tall. A person 1.8m tall walks away from the lamppost at 1 m/s. What is the rate at which the length of the shadow is growing?

In the same situation as the previous exercise: what is the speed of the shadow's tip (distance from the lamppost)?

A rectangular prism reservoir has a base width $b = 4$ m and length $L = 10$ m. If the height $h$ is increasing at 0.1 m/s, what is $dV/dt$?

A right triangle has legs $a = 3$ cm and $b = 4$ cm. Leg $a$ is growing at 1 cm/s; $b$ is fixed. What is the rate of change of the hypotenuse?

An airplane flies horizontally at 500 km/h, 5 km above an observer. What is the rate of change of the distance between the plane and the observer 1 minute after the plane passes overhead?

A boat is pulled toward a dock 6 m above the water by a rope of length 10 m. The rope is being pulled in at 1 m/s. At what speed is the boat approaching the dock (horizontally)?

Car A travels north at 50 km/h; Car B travels east at 60 km/h. What is their rate of separation after 30 min of travel?

A TV camera is 30m from a racetrack. A car passes at 80 m/s. What is the angular rate of the camera when the car is directly in front of it?

A snowball melts such that $dV/dt = -k \cdot A$ where $A = 4\pi r^2$ is the surface area. Show that $dr/dt = -k$ (constant).

An equilateral triangle has side length $a$ growing at 1 cm/s. What is $dA/dt$ when $a = 10$ cm?

Why is it an error to substitute the numerical value of a variable before differentiating the equation with respect to $t$? Choose the most precise explanation.

What differentiation rule is the mathematical foundation of related rates?

In the sliding ladder problem, the foot moves away from the wall ($\dot x > 0$). Prove that $\dot y < 0$ whenever $x, y > 0$.

In a conical tank filling at a constant rate, when does the water level rise most rapidly?

In the conical tank, $V = \frac{1}{3}\pi r^2 h$ depends on two variables ($r$ and $h$). Explain the procedure to eliminate the extra variable before differentiating.

When differentiating $\tan\theta = h/x$ with respect to $t$, what factor appears multiplying $d\theta/dt$ on the left side?

For a circle with radius increasing at a constant rate $dr/dt = c$, how does $dA/dt$ behave as $r$ increases? Justify.

What distinguishes related rates problems from one another (balloon, ladder, tank, shadow)?

A camera tracks an object moving past it at constant speed. At what instant is the camera rotating fastest? Justify algebraically.

Differentiate $V = \frac{4}{3}\pi r^3$ with respect to $t$ and explain why the resulting coefficient $4\pi r^2$ has geometric significance.

In the SIR model, $\dot S = -\beta SI$ with $\beta = 0.001$, $S_0 = 999$, $I_0 = 1$. What is $\dot S$ at the initial instant?

Chemical reaction $A \to B$ with $\dot A = -kA$. Determine the half-life of $A$ in terms of $k$.

In the Verhulst logistic model $\dot P = rP(1 - P/K)$, at what value of $P$ is the growth rate $\dot P$ maximized?

A cylindrical tank of radius $R$ has an orifice of area $A_s$ at the bottom. By Torricelli's law, the outflow velocity is $\sqrt{2gh}$. Derive the ODE for $dh/dt$.

A cylinder: radius grows at 1 cm/s, height $h = 20$ cm is constant. What is $dV/dt$ when $r = 5$ cm?

A plane at 800m altitude flies horizontally at 200 m/s toward an observer. What is the rate of change of the angle of elevation when the plane is 600m horizontally from the observer?

A lamppost of height $h$, a person of height $p$ walking away from the post at speed $v$. Derive the general formula for the speed of the tip of the shadow.

Proof. Prove rigorously that $V = \frac{4}{3}\pi r^3$ implies $\dfrac{dV}{dt} = 4\pi r^2\dfrac{dr}{dt}$, showing each step of the chain rule application. Interpret the factor $4\pi r^2$ geometrically.

Proof. For the sliding ladder with $x^2 + y^2 = L^2$, show rigorously that $\dot x$ and $\dot y$ always have opposite signs when $x, y > 0$.

Proof. A camera tracks an object moving along a straight line at a distance $d$ (perpendicular). Derive the general formula for $d\theta/dt$ in terms of $\theta$, $d$, and $dx/dt$. Identify when the rotation is maximum.