Why does water rise more slowly as a cone fills?
You will be able to: Use similar triangles to eliminate a changing radius and find a height rate.
BC foundation: Unit 4 shares these contextual differentiation objectives with AB. Interpret signed rates with units, relate changing quantities before substituting an instant, and check the conditions behind approximations and limit methods. Parametric and polar motion come later.
Why does water rise more slowly as a cone fills?
Water enters an upright cone with its tip at the bottom. Near the tip, a little added water makes the level rise quickly. Higher up, the same volume must cover a wider surface.
A useful starting point: How fast does the top of a sliding ladder move? →
Words and symbols before equations
- h(t)
- Water depth measured upward from the cone’s tip, in meters.
- r(t)
- Radius of the water’s horizontal surface, in meters.
- Similar triangles
- Cross-sections with matching angles and equal corresponding side ratios.
- Q=V′
- Net volume rate in cubic meters per minute, signed positive for filling.
What this picture assumes
Original mathematical model; readouts are rounded. Tip-down cone: fixed H=6 m,R=3 m; water r=h/2 and V=πh³/12. Q=qπ m³/min; negative q means draining. Controls are snapshots, not a time simulation. Formula only for 0<h<6. 3D and 2D use the same geometry; camera rotation changes no values.
Read the picture in three steps
- Read the axes and labels first. Identify what each symbol and line represents. Read the units and fixed conditions before comparing quantities.
- h=4 m, r=2 m; V=16.7552 m³. Net flow Q=2π=6.28319 m³/min; water-surface area=12.5664 m². h′=Q/area=0.5 m/min; r′=0.25 m/min. Filling.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
The tank is 6 m high with top radius 3 m. Its axial cross-section gives r/h=3/6, so r=h/2 at every water level. The tank dimensions are fixed, but the water radius changes.
Substitute that relation into V=(1/3)πr²h to obtain V=πh³/12. Now only h varies: V′=(πh²/4)h′.
At h=4 m and Q=2π m³/min, h′=4Q/(πh²)=0.5 m/min. The water surface has area πr²=4π m², and dividing volume rate by this area gives the same height rate.
The optional 3D cone shows the circular surface and the axial triangle that fixes r/h. Its labels, 2D cross-section and equations carry the full explanation. The formula applies for 0<h<6; the model does not divide by zero at an empty tip or continue beyond an overflowing rim.
A worked example, step by step
Find the water-height rate at h=2 m if this tank fills at π m³/min.
- Similar triangles give r=h/2, so r=1 m at this instant.
- V=πh³/12 gives V′=(πh²/4)h′.
- Substitute π=(π·4/4)h′.
- Thus h′=1 m/min upward; the same flow would cause a smaller height rate at a greater depth.
The tank’s top radius 3 m is not the water-surface radius at every level. Treating that changing radius as fixed gives the wrong volume relation.
At fixed Q, does doubling h halve h′?
Compare with an explanation
No. h′ is proportional to 1/h², so doubling h makes h′ one quarter as large.
Predict. Change one thing. Explain.
Keep Q fixed and compare depths 2 and 4 m. Inspect the water surface in 3D or the labeled cross-section. Explain why doubling depth quarters the height rate.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
h=4 m, r=2 m; V=16.7552 m³. Net flow Q=2π=6.28319 m³/min; water-surface area=12.5664 m². h′=Q/area=0.5 m/min; r′=0.25 m/min. Filling.
Original mathematical model; readouts are rounded. Tip-down cone: fixed H=6 m,R=3 m; water r=h/2 and V=πh³/12. Q=qπ m³/min; negative q means draining. Controls are snapshots, not a time simulation. Formula only for 0<h<6. 3D and 2D use the same geometry; camera rotation changes no values.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using the relevant rate relationship, tangent estimate, or limit argument and its conditions. Identify what the representation cannot tell you.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionFor the same 6 m by 3 m tank, calculate h′ and r′ when h=3 m and Q=3π m³/min.
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Compare with the answer and four-point rubric
- 1 point: The water radius is r=1.5 m.
- 1 point: Q=(πh²/4)h′ gives h′=4(3π)/(9π)=4/3 m/min.
- 1 point: Differentiate r=h/2 to obtain r′=h′/2.
- 1 point: r′=2/3 m/min; both dimensions grow while preserving r/h=1/2.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1Why eliminate r before differentiating?
Similar triangles express it in terms of h and avoid an extra unknown rate.
RECALL 2What physical area divides the volume rate?
The horizontal water surface area πr².
RECALL 3Does rotating the camera alter Q or h′?
No; it changes only how the same geometry is viewed.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Why does water rise more slowly as a cone fills?
- r/h=R/H=1/2; V=πh³/12.
- h′=4Q/(πh²)=Q/(πr²), 0<h<6.
Remember: The tank’s top radius 3 m is not the water-surface radius at every level. Treating that changing radius as fixed gives the wrong volume relation.
Conditions: Original mathematical model; readouts are rounded. Tip-down cone: fixed H=6 m,R=3 m; water r=h/2 and V=πh³/12. Q=qπ m³/min; negative q means draining. Controls are snapshots, not a time simulation. Formula only for 0<h<6. 3D and 2D use the same geometry; camera rotation changes no values.
Refresh Kid · AP Calculus BC Unit 4 · Objectives CHA-3.E · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 4.5, CHA-3.E. CED effective Fall 2020, with Fall 2026 clarifications, checked September 17, 2026. Unit 4 has seven official topics. Topic 4.7 assesses 0/0 and ∞/∞ quotient forms; other indeterminate forms are excluded from the core lessons. Focused lesson titles and questions are original Refresh Kid teaching material.
Derivative units and signs are interpreted in context. Speed is the magnitude of velocity; turning requires a sign change. Related-rate equations hold at nearby times and are differentiated before snapshot values are inserted. Cone and ladder models have explicit physical domains. Tangent approximations remain estimates; error direction requires behavior on the relevant interval. L’Hôpital’s rule requires an eligible quotient form, nearby differentiability, nonzero denominator derivative and an existing finite or infinite derivative-ratio limit. A failed derivative-ratio limit is inconclusive about the original quotient.
The Organic Chemistry Tutor video creators and relevant descriptions were checked; full videos were not reviewed. Khan Academy’s unit destination was checked; its JavaScript lesson content was not fully readable by the research tool. OpenStax Sections 3.4, 4.1, 4.2 and 4.8 were consulted for conceptual cross-checking. No creator scripts, questions, diagrams or artwork were copied. Refresh Kid is not endorsed by these providers.
GitHub’s 3D website collection and its camera-control example informed optional spatial inspection. Graph geometry is generated from the stated original equations; self-hosted Three.js retains its MIT license. The optional 3D model shows the circular water surface and axial cross-section of a tip-down conical tank. The water radius and height obey the same similar-triangle ratio in 2D and 3D. Camera rotation only changes the view; signed flow and height controls describe an instantaneous state. Complete labeled 2D geometry, rates and equations remain available without WebGL.
Independent teacher review and observation of students remain pending. Checks do not certify scientific accuracy, accessibility or learning effectiveness. This is a review edition.
Optional official resource: Released AP Calculus BC questions and scoring guides. The archive spans multiple units; no entire exam question is assigned to this lesson.
Learn → Explore → Practice → Review is informed by the IES learning guide; this exact implementation has not been evaluated with learners.
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