Related Rates

Main idea

If an equation ties several quantities together and they're all changing over time, their rates of change are tied together too. Differentiate the equation with respect to $t$ , and you can find the rate you don't know from the ones you do.

Example: a growing circle

A = \pi r^2 \implies {\htmlData{sym=1}{\frac{dA}{dt}}} = 2\pi r\,{\htmlData{sym=0}{\frac{dr}{dt}}}

What the symbols mean

$\frac{dr}{dt}$
Known rate
How fast the radius grows.
$\frac{dA}{dt}$
Unknown rate
How fast the area grows as a result.

Explanation

A related rates problem is simply one where two or more things are changing at once, and because they're connected, knowing how fast one changes tells you how fast the other does. Think of blowing up a balloon: you control how fast the air goes in, and that decides how fast the radius grows. The formula $\frac{dA}{dt} = 2\pi r\frac{dr}{dt}$ looks technical, but it's just $\frac{dr}{dt}$ (how fast the radius is growing, which you know), $\frac{dA}{dt}$ (how fast the area is growing, which you want) and $2\pi r$ (the link between them, from differentiating $A = \pi r^2$ ). It's implicit differentiation again, with time as the variable.

Last lesson, $y$ depended on $x$ and every $y$ term picked up a $y'$ . Here, everything depends on time $t$ . So when you differentiate an equation like $x^2 + y^2 = 100$ with respect to $t$ , every variable picks up its own rate: $2x\,\frac{dx}{dt} + 2y\,\frac{dy}{dt} = 0$ . That one equation links the two rates, so if you know one, you can solve for the other.

Steps:

Draw a picture and name every changing quantity.
Write an equation linking them. This is usually geometry: Pythagoras, similar triangles, area or volume formulas.
Differentiate both sides with respect to $t$ . Every variable gets a rate via the chain rule.
Only now plug in the values at the instant you care about, and solve for the unknown rate.

A common mistake. It's tempting to plug in the numbers right away, since you know them. Say the ladder's foot is 6 ft from the wall, so you write $x = 6$ before differentiating. But then $x$ is a constant, its derivative is 0, and you've thrown away the very rate you were given. A quantity that's changing has to stay a variable until after you take the derivative. The numbers only describe one instant, and the derivative needs to see how things change around it.

Signs carry meaning too: a negative rate means the quantity is shrinking.

See it

A 13 ft ladder's foot slides away from the wall at a steady 2 ft/s. Because x² + y² = 169, the top has to fall at 2x/y ft/s. Notice how uneven that is: under 1 ft/s at the start, when the foot is 5 ft out, but nearly 5 ft/s by the time the foot reaches 12 ft and the ladder is almost flat.

Worked examples

Example 1

A 10 ft ladder leans on a wall. Its bottom slides away at 1 ft/s. How fast is the top sliding down when the bottom is 6 ft from the wall?

1
Let $x$ be the bottom's distance from the wall and $y$ the top's height. The ladder, wall and ground make a right triangle, so $x^2 + y^2 = 100$ .

Example 2

Air is pumped into a spherical balloon at 100 cm³/s. How fast is the radius growing when $r = 5$ cm?

1
Volume and radius are linked by $V = \frac43\pi r^3$ . Differentiating with respect to time gives $V' = 4\pi r^2 r'$ .

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

A circle's radius grows at 2 cm/s. How fast is its area growing when $r = 10$ cm?

Problem 2+5 XP

A 13 ft ladder's bottom is pulled away at 2 ft/s. How fast is the top moving when the bottom is 5 ft from the wall? (Negative means down.)

Problem 3+5 XP

A cube's edge grows at 0.1 cm/s. How fast is its volume growing when the edge is 3 cm?

Problem 4+5 XP

A sphere's volume grows at $36\pi$ m³/min. How fast is the radius growing when $r = 3$ m?

Problem 5+5 XP

Water pours at 9 ft³/min into a cone (point down) 10 ft tall with a 5 ft top radius. How fast is the water rising when it's 6 ft deep?

Problem 6+5 XP

A drone rises straight up at $3$ m/s from a spot $40$ m from an observer. How fast is the distance between them growing when the drone is $30$ m up?

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

Water pours into a conical tank (point down, $10$ m tall, top radius $4$ m) at $2$ m³/min, while a leak at the bottom drains $0.1h$ m³/min when the water is $h$ m deep. How fast is the depth changing when $h = 5$ m?

Where it's used

Radar tracking (how fast is the distance to a plane changing?), inflating airbags, draining reservoirs and spreading oil slicks are all related-rates problems.

Tracking a Car with a Drone CameraCalc I · Robotics

Implicit Differentiation Higher Order Derivatives