Related Rates Problem Setup

About This MicroSim

Most related rates mistakes happen before any calculus is done. They happen in the diagram: a number gets written where a letter belongs, a rate gets the wrong sign, or the wrong equation gets chosen. This MicroSim shows two classic problems side by side, the sliding ladder and the inflating balloon, and builds the labeled diagram for each one in five steps.

Step	What you add to the diagram	Color	Why
1	A clean sketch		You can't label what you haven't drawn
2	Letters for everything that changes (\(b\), \(h\), \(r\), \(V\))	Blue	A quantity that changes with time must stay a variable
3	Numbers for everything that stays constant (\(L = 10\) ft)	Green	Only a true constant may be written as a number
4	Arrows for each rate (\(\frac{db}{dt}\), \(\frac{dh}{dt}\), \(\frac{dr}{dt}\), \(\frac{dV}{dt}\))	Red	The arrow direction tells you the sign of the rate
5	The equation linking the labels	Purple	This is what you differentiate with respect to \(t\)

The two relating equations are:

\[b^2 + h^2 = L^2 \qquad \text{and} \qquad V = \frac{4}{3}\pi r^3\]

The Time slider is the test for step 2 and step 3. Drag it and watch the readouts under each picture. Anything whose value changes (\(b\), \(h\), \(r\), \(V\)) needs a letter. The one value that refuses to budge (\(L = 10\) ft) is the constant.

Delta Moment

"Before I roll anywhere, I look around and ask: what's moving and what's standing still? The moving stuff gets a letter. The stuck stuff gets a number. Mix those up and the whole trip goes sideways!"

How to Use

The MicroSim opens on the finished diagram (step 5), so you can see where you're headed.
Click Start Over to clear the labels and return to the bare sketch.
Before each click of Next, predict what will be added. Which quantities deserve a letter? Which rate is negative?
Read the yellow callout. It explains what was just labeled and why.
Drag the Time slider at any step to slide the ladder and inflate the balloon. Watch which readouts change.
Use Previous to step back and compare two stages of the diagram.

What to Observe

In the ladder problem, \(b\) grows while \(h\) shrinks, so \(\frac{db}{dt} > 0\) and \(\frac{dh}{dt} < 0\). The arrows point right and down.
In the balloon problem, every arrow points outward. Both \(r\) and \(V\) are growing, so both rates are positive.
\(L\) stays at 10 ft for every position of the slider. That is why it is the only quantity written as a number.
The values \(b = 6\) ft and \(h = 8\) ft appear at exactly one instant (\(t = 2\) sec). They are not constants, so they get substituted only after differentiating.

Iframe Embed Code

You can add this MicroSim to any web page by adding this to your HTML:

<iframe src="https://dmccreary.github.io/calculus/sims/related-rates-problem-setup/main.html"
        height="567px"
        width="100%"
        scrolling="no"></iframe>

Lesson Plan

Grade Level

Grades 11-12 (college placement Calculus)

Duration

10-15 minutes

Learning Objectives

After using this MicroSim, students will be able to:

Interpret a related rates scenario by identifying which quantities change with time and which stay constant (Bloom Level 2)
Explain why a changing quantity must be labeled with a variable instead of its value at one instant
Classify each rate in a diagram as positive or negative from the direction of its arrow
Identify the equation that relates the labeled quantities

Prerequisites

The Pythagorean theorem and the volume formula for a sphere
Derivative as a rate of change, including the notation \(\frac{dy}{dt}\)
Implicit differentiation and the chain rule

Activities

Predict, then reveal (5 min): Click Start Over. Before each click of Next, students write down what they think will be added to both diagrams. Compare predictions with the reveal.
The constant test (3 min): At step 3, drag the Time slider from 0 to 3 seconds. Students list every readout that changed and every readout that did not. Ask: why is it a mistake to write "8 ft" on the wall in the diagram?
Sign check (3 min): At step 4, students explain in one sentence why \(\frac{dh}{dt}\) is negative but \(\frac{db}{dt}\) is positive, using the arrows as evidence.
Transfer (4 min): Students sketch and label a new scenario (for example, a circular ripple spreading on a pond) using the same four colors: variables, constants, rates, equation.

Discussion Questions

The problem says "the base is 6 feet from the wall." Why do we still label the base \(b\) and not 6?
The balloon panel has no green label. What does that tell you about the balloon problem?
If the ladder were being pushed toward the wall, which arrows would flip? Which signs would change?
Why must the equation be written before any numbers are substituted?

Assessment

Students have met the objective when they can take an unfamiliar related rates problem and produce a diagram that:

labels every changing quantity with a variable
labels only true constants with numbers
shows each given or unknown rate with an arrow and the correct sign
states one equation that relates the labeled variables