How large should the corner cuts be to maximize a box’s volume?
You will be able to: Connect a flat net to a three-dimensional box and justify its maximum volume.
BC foundation: Unit 5 shares these analytical differentiation objectives with AB. Check theorem hypotheses, justify derivative signs over intervals, and compare feasible candidates before claiming an optimum. State the local branch when analyzing an implicit relation.
How large should the corner cuts be to maximize a box’s volume?
Cut equal squares from the corners of a 12 cm by 12 cm sheet, then fold the remaining flaps upward. Larger cuts make taller sides but a smaller base, so taller does not always mean more volume.
A useful starting point: How do you prove a garden design has the greatest area? →
Words and symbols before equations
- Cut size x
- The side length of each removed corner square, in cm.
- Box net
- The remaining flat base and four flaps before folding.
- Base side 12−2x
- The sheet length after losing a cut at both ends.
- Volume V
- Base area multiplied by height, in cm³.
What this picture assumes
Original model; numerical readouts are rounded. 12×12 cm sheet; cut four x×x corners. Completed box base=(12−2x)², height=x, 0<x<6. Fold angle is a construction view only: volume always refers to the completed upright box. Negligible thickness, no tabs or overlap. 3D and net use the same lengths.
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.
- Cut x=2 cm; completed box 8×8×2 cm. Volume=128 cm³ and V′=0 cm³/cm. Fold view=90°; only 90° is upright. The volume always describes the upright design. This is the maximum: 128 cm³.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
After folding, height is x and both base sides are 12−2x. Hence V(x)=x(12−2x)² for 0<x<6. The optional 3D view shows the same net folding; fold angle is only a construction view, and the volume formula describes the fully upright box.
Expand V=144x−48x²+4x³, then differentiate: V′=144−96x+12x²=12(x−2)(x−6). Its only interior zero is x=2; x=6 would collapse the base.
On (0,2), both factors are negative and V′>0. On (2,6), the factors have opposite signs and V′<0. This proves the global maximum at x=2; the limiting volume is zero at either boundary.
The resulting box has 8 cm by 8 cm base, height 2 cm and volume 128 cm³. We assume negligible sheet thickness and no tabs or material overlap; actual manufacturing would need a different constraint.
A worked example, step by step
For a 9 cm square sheet with the same corner-cut design, find the maximizing cut and volume.
- V=x(9−2x)² on 0<x<4.5.
- V′=(9−2x)(9−6x), with interior zero x=1.5.
- The derivative is positive before 1.5 and negative after within the domain.
- The box has base 6 cm by 6 cm, height 1.5 cm and maximum volume 54 cm³.
Each base dimension loses 2x, not x. Reject the collapsed boundary root, and do not confuse an intermediate fold with the completed box.
Does rotating or folding the display change the chosen cut size?
Compare with an explanation
No. These are viewing/construction controls; the cut-size control sets the design and the completed-box volume.
Predict. Change one thing. Explain.
Change the corner cut, then fold from flat to upright in 3D or inspect the 2D net. Explain why increased height can be outweighed by lost base area. Check x=1,2 and 3.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Cut x=2 cm; completed box 8×8×2 cm. Volume=128 cm³ and V′=0 cm³/cm. Fold view=90°; only 90° is upright. The volume always describes the upright design. This is the maximum: 128 cm³.
Original model; numerical readouts are rounded. 12×12 cm sheet; cut four x×x corners. Completed box base=(12−2x)², height=x, 0<x<6. Fold angle is a construction view only: volume always refers to the completed upright box. Negligible thickness, no tabs or overlap. 3D and net use the same lengths.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using the relevant derivative signs, theorem conditions, domain or geometric constraint. 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 a 6 cm square sheet, find the corner cut maximizing the open-box volume and justify it over the feasible domain.
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Compare with the answer and four-point rubric
- 1 point: V=x(6−2x)² for 0<x<3.
- 1 point: V′=(6−2x)(6−6x), with sole interior zero x=1.
- 1 point: Its sign changes positive to negative across the feasible domain.
- 1 point: Base 4 cm by 4 cm, height 1 cm, maximum volume 16 cm³.
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 subtract 2x from a base side?
One cut is removed at each end.
RECALL 2What is the feasible domain for the 12 cm sheet?
0<x<6; both height and base dimensions must be positive.
RECALL 3What does the fold angle represent?
The construction from net to upright walls, not an extra optimization variable.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How large should the corner cuts be to maximize a box’s volume?
- V=x(12−2x)², 0<x<6.
- V′=12(x−2)(x−6); maximum at x=2 with V=128 cm³.
Remember: Each base dimension loses 2x, not x. Reject the collapsed boundary root, and do not confuse an intermediate fold with the completed box.
Conditions: Original model; numerical readouts are rounded. 12×12 cm sheet; cut four x×x corners. Completed box base=(12−2x)², height=x, 0<x<6. Fold angle is a construction view only: volume always refers to the completed upright box. Negligible thickness, no tabs or overlap. 3D and net use the same lengths.
Refresh Kid · AP Calculus BC Unit 5 · Objectives FUN-4.C · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 5.11, FUN-4.C. CED effective Fall 2020, with Fall 2026 clarifications, checked September 17, 2026. Unit 5 has twelve official topics. Focused lesson titles and questions are original Refresh Kid teaching material.
Existence theorems require their stated hypotheses. Critical numbers must belong to the function’s domain; interior local extrema and included endpoints are handled explicitly. Extrema and inflection candidates need justification, not just a zero derivative. Sparse derivative samples do not establish interval-wide signs. Optimization includes the constraint, feasible domain and global comparison. Implicit-curve conclusions specify the branch and distinguish finite slopes from vertical tangents.
The Organic Chemistry Tutor Mean Value Theorem and Optimization Problems video titles, creator and relevant descriptions were checked; full videos were not reviewed. The free optimization video mentions additional paid material, which is not required here. Khan Academy’s unit destination was checked; its JavaScript lesson contents were not fully readable by the research tool. OpenStax Sections 4.3, 4.4, 4.5 and 4.7 were consulted for conceptual cross-checking. No creator scripts, questions, diagrams or artwork were copied. Refresh Kid is not affiliated with or endorsed by these providers.
GitHub’s 3D website collection and its camera-control example informed optional spatial inspection. The open-box model is original geometry using existing self-hosted Three.js with its MIT license. A 12 cm square sheet loses four equal corner squares. The remaining net folds into an open box; the geometry uses the same lengths as the labeled 2D net. Fold angle is a construction view, not an independent design variable; displayed volume refers to the fully upright box. Camera rotation changes no mathematical values. Complete 2D diagrams and explanations 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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