4.3 Related Ratesap Calculus

4.4 Introduction to Related Rates Calculus Find a relationship between the given rates of change by doing the following. Set up a relationship using variables for the situation. Differentiate with respect to time 𝒕. An ice cube is melting. Relate the volume’s rate of change with the edges’ rate of change. This type of problem is known as a 'related rate' problem. In this sort of problem, we know the rate of change of one variable (in this case, the radius) and need to find the rate of change of another variable (in this case, the volume), at a certain point in time (in this case, when r = 4). The reason why such a problem can be solved is that.


This video lesson explores the concept of Related Rates, which is the study of what is happening over time.

4.3 Related Ratesap Calculus

To solve problems with Related Rates, we will need to know how to differentiate implicitly, as most problems will be formulas of one or more variables.

But this time we are going to take the derivative with respect to time, t, so this means we will multiply by a differential for the derivative of every variable!

Calculus Related Rates Problem: At what rate does the angle change as a ladder slides away from a house? A 10-ft ladder leans against a house on flat ground. The house is to the left of the ladder. The base of the ladder starts to slide away from the house at 2 ft/s.

But don’t let this confused you, as there are only four basic steps when problem-solving with Related Rates:

  1. Write down every given information or quantity, even what you’re looking for, and make a sketch.
  2. Determine the equation needed to solve for your unknown quantity.
  3. Take the derivative implicitly, not forgetting to multiply by your differential for every single variable!
  4. Plug in all of your given information or quantities and solve.

And here’s a big hint…

…most of the time all we have to do is find an equation that relates the rate we’re looking for to a rate that we already know, as Paul’s Online Notes so nicely states.

This means we need to look for geometric shapes, known formulas, and ratios.

Ladder Sliding Down a Wall

Together we will solve these 8 classic questions:

4.3 related ratesap calculus calculator
  • Pebble Dropped In Pond
  • Air Being Pumped Into A Balloon
  • Edges Of Cube Expanding
  • Radar Tracking Station & Airplane Flight Path
  • Sand Falling onto a Conical Pile
  • Water Pouring into a Conical Tank
  • Boat Pulled into a Dock
  • Ladder Sliding Down a Wall

How to Solve Related Rates – Video

1 hr 35 min

  • Overview of Related Rates + Tips to Solve Them
  • 00:02:58 – Increasing Area of a Circle
  • 00:12:30 – Expanding Volume of a Sphere
  • 00:21:15 – Expanding Volume of a Cube
  • 00:26:32 – Calculate the Speed of an Airplane
  • 00:39:13 – Conical Sand Pile
  • 00:51:19 – Conical Water Tank
  • 00:59:59 – Boat & Winch
  • 01:09:13 – Ladder Sliding Down A Wall

Related Rates

4.3 Related Ratesap Calculus

Related Rates – 2 Examples

4.3 Related Ratesap Calculus

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