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Teaching video
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Grade 12 chapters and video availability01 · Read and understand
What you will learn
- Use geometric similarity before differentiating a related-rates model.
- Justify the conclusion "At h=2 cm, dh/dt=1/π cm/s" using the stated assumptions.
Before you start
Cone volume, similar triangles, and derivatives.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
In a conical container the liquid surface radius satisfies r=h/2. With lengths in centimeters, if dV/dt=1 cm³/s, find dh/dt when h=2 cm.
Why this math matters
Use geometric similarity before differentiating a related-rates model. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The ideal container keeps the stated cone proportions.
- Units are chosen consistently and no volume is lost.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Connect volume change to a changing height
PausedQuestion: Start with the question. Paused.
Question
Start with the question
In a conical container the liquid surface radius satisfies r=h/2. With lengths in centimeters, if dV/dt=1 cm³/s, find dh/dt when h=2 cm.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
V=πr²h/3=πh³/12
Similarity removes r so the volume depends on one changing dimension.
Work through the mathematics
dV/dt=(πh²/4)dh/dt
Differentiate with respect to time, retaining the chain factor.
Check the conclusion
At h=2 cm, dh/dt=1/π cm/s
The same added volume produces different height changes at different fill depths.
The result
At h=2 cm, dh/dt=1/π cm/s
The same added volume produces different height changes at different fill depths.
Common mistakes to catch
- Substitute a geometric relation before eliminating a changing variable.
- Insert the instant's height after differentiating the variable model.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
What is dh/dt when h=1 cm with the same flow?
Show a hint
Substitute into 1=(πh²/4)h′.
Reveal answer and explanation
4/π cm/s
The narrower cross section raises the level faster.
Practice 2
Why not hold r constant while differentiating?
Show a hint
r changes with h in this shape.
Reveal answer and explanation
It would violate the similarity relation
Both geometric variables depend on the liquid depth.
Take the idea with you
Compare equal volume inflows in containers with changing versus constant cross-sectional area.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Guarantee a horizontal tangent between equal endpoint heights
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