A-level · Beta
Applied optimisation
Cut and fold an open box; connect its greatest volume to a zero derivative.
Change the sheet dimensions and corner cut. The net and volume graph update together.
A slightly larger cut increases volume here.
- Box base (cm)
- 24 × 14
- Height (cm)
- 3
- Volume (cm³)
- 1008
- dV/dx (cm²)
- 108
- Best cut (cm)
- 3.9237
- Maximum volume (cm³)
- 1056.3
Use 0 < x < half the shorter sheet side for a physical box. Endpoints show limiting zero-volume cases. Coordinates and volumes are rounded.
Predict, test and explain
Step 1 of 3
Predict
Does the largest possible corner cut give the largest box volume?
Explore and compare
Choose the investigation above. Predict what will change before you move a control.
Try it in the simulation. Change one thing at a time.
Notes and saved values stay in this tab. They disappear when you reload.
Hints, self-check and connections
Try a self-check
Fixed values, separate from the diagram controls.
Does the largest possible corner cut give the largest box volume?
Hint
Write volume in terms of one variable and check its physical domain.
Reveal answer
Increasing the cut raises the height but shrinks both base dimensions. The boundary has zero volume.
Build an explanation
Hint 1
Write volume in terms of one variable and check its physical domain.
Hint 2
Increasing the cut raises the height but shrinks both base dimensions. The boundary has zero volume.
Hint 3
Use the model to check a prediction, then explain the result.
Worked example
This example uses fixed values, separate from the diagram controls.
Find the best cut for an open box made from a 12 cm by 12 cm sheet.
- V = x(12 − 2x)² for 0 < x < 6.
- V′ = 144 − 96x + 12x²; solve V′ = 0.
- The interior root is x = 2; x = 6 is a degenerate boundary. The derivative changes from positive to negative at 2.
Answer: Cut 2 cm; maximum volume 128 cm³.
Connect this idea
Watch out: Increasing the cut raises the height but shrinks both base dimensions. The boundary has zero volume.
Specification and learning route
AQA G3 · Edexcel Pure 7.3
AS and A-level. This model illustrates selected content; references are not a claim of full coverage or exam-board endorsement.
Useful starting knowledge: Differentiation; Stationary points; Forming algebraic constraints; Geometry or context formulas.