How much remains when material enters and leaves?
You will be able to: Build an amount function and justify an absolute maximum from net rate.
How much remains when material enters and leaves?
A tank receives water while a drain runs. A fast inlet does not guarantee the amount is increasing: compare inflow with outflow first.
A useful starting point: How do initial values connect acceleration, velocity and position? →
Words and symbols before equations
- Inflow I
- Nonnegative rate entering, in L/min.
- Outflow O
- Nonnegative rate leaving.
- Net rate
- I−O, the derivative of stored amount.
- Candidate time
- Endpoint or interior time where the net rate is zero or undefined.
What this picture assumes
Original model; numerical labels are rounded. I=4−t L/min, O=1 L/min, initial amount 5 L; Q=5+3t−t²/2. Model assumes tank capacity at least 10 L and no other flows; amount stays positive.
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.
- t=3 min; inlet 1, outlet 1 L/min; net 0 L/min; amount 9.5 L. Absolute maximum 9.5 L at t=3 min.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
Let I(t)=4−t and O(t)=1 on [0,4] minutes, with Q(0)=5 L. The net rate is 3−t, so Q(t)=5+3t−t²/2.
The amount increases before 3 and decreases afterward. The rates are equal at 3, making it a candidate maximum; this says nothing about equality of total entered and total left.
Compare Q(0)=5, Q(3)=9.5 and Q(4)=9. The absolute maximum is 9.5 L at 3 minutes. State the amount and its time, and justify why all candidates were considered.
A physical model also requires Q≥0 and no unintended overflow. This example stays between 5 and 9.5 L in a tank of capacity at least 10 L.
A worked example, step by step
A tank starts with 10 L, receives 5 L/min and drains at 1+t L/min on [0,5]. Find its maximum amount.
- Net rate is 5−(1+t)=4−t.
- Q=10+4t−t²/2, with critical time 4.
- Compare Q(0)=10, Q(4)=18 and Q(5)=17.5.
- The maximum is 18 L at 4 min, assuming capacity is at least 18 L.
Maximize the amount Q, not the inflow rate I. Equal rates indicate zero amount derivative, not an empty tank.
When I=O, must Q=0?
Compare with an explanation
No. Equal rates mean the amount is momentarily stationary, regardless of its height.
Predict. Change one thing. Explain.
Move time across 3 minutes. Compare inlet, outlet and stored amount. Explain why Q is still positive after its derivative becomes negative.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
t=3 min; inlet 1, outlet 1 L/min; net 0 L/min; amount 9.5 L. Absolute maximum 9.5 L at t=3 min.
Original model; numerical labels are rounded. I=4−t L/min, O=1 L/min, initial amount 5 L; Q=5+3t−t²/2. Model assumes tank capacity at least 10 L and no other flows; amount stays positive.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using the units, signed change, strip direction, radius distances or cross-sectional area. 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 Q(0)=8, I=6−t and O=2 on [0,5], find Q(5) and the absolute maximum.
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Compare with the answer and four-point rubric
- 1 point: Q′=4−t and Q=8+4t−t²/2.
- 1 point: Q(5)=15.5.
- 1 point: Candidates are 0,4,5 with values 8,16,15.5.
- 1 point: Maximum 16 at t=4; the net rate changes + to −.
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 1Which rate must be integrated?
Inflow minus outflow.
RECALL 2What does zero net rate mean?
The amount has zero instantaneous rate of change.
RECALL 3Why check endpoints?
Absolute extrema can occur there.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How much remains when material enters and leaves?
- Q(t)=Q(a)+∫ₐᵗ(I−O)du.
- Check endpoints and net-rate critical times.
- Physical amounts must satisfy the model’s capacity and nonnegativity conditions.
Remember: Maximize the amount Q, not the inflow rate I. Equal rates indicate zero amount derivative, not an empty tank.
Conditions: Original model; numerical labels are rounded. I=4−t L/min, O=1 L/min, initial amount 5 L; Q=5+3t−t²/2. Model assumes tank capacity at least 10 L and no other flows; amount stays positive.
Refresh Kid · AP Calculus BC Unit 8 · Objectives CHA-4.D, CHA-4.E · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 8.3, CHA-4.D, CHA-4.E. CED effective Fall 2020 and Fall 2026 clarifications checked September 17, 2026. Unit 8 covers BC topics 8.1–8.13, including smooth planar graph arc length and distance traveled. Focused lesson titles, examples, questions and illustrations are original Refresh Kid material. Unit 7 differential equations is available separately. Parametric and polar curves belong to later units.
Average value is distinguished from average rate. Motion uses velocity for displacement and speed for distance, with initial values stated separately. Area bounds and ordering are checked, including multiple crossings. Volumes derive the slice area before integration, distinguish diameter from radius, and use distances from the specified axis. Disks and washers use perpendicular slices and a consistent integration variable. A region crossing an axis requires checking the actual swept disk rather than inventing a hole.
The Organic Chemistry Tutor Disk & Washer Method and Arc Length Calculus Problems video titles and creator were checked; the full video was not reviewed. Use the free video as an optional companion; no paid material is required. Khan Academy’s unit destination was checked, but its JavaScript lesson contents were not fully readable by the research tool. OpenStax Sections 6.1 and 6.2 and the arc-length subsection of Volume 2 Section 2.4 were consulted for conceptual cross-checking. No provider scripts, questions, artwork or diagrams were copied. Refresh Kid is not affiliated with or endorsed by these providers.
GitHub’s 3D website collection informed optional spatial inspection and camera controls. All cross-section and revolution geometry is original and uses existing self-hosted Three.js with its MIT license. Spatial coordinates preserve the mathematical lengths; sampled mesh surfaces illustrate exact formulas. Teal shows the solid and orange a selected zero-thickness section. The volume is for the entire solid, not the highlighted plane. No autoplay is used; camera rotation changes only the view. Labeled 2D regions, cross-section diagrams, readouts and calculations remain available without WebGL.
Independent teacher review and observation of students remain pending. Technical checks do not certify mathematical 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 here.
Learn → Explore → Practice → Review is informed by the IES learning guide; this exact implementation has not been evaluated with learners.
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