Review: Build and Use a Quadratic Area Model
Create a constrained area rule, inspect its changes, find a maximum, and solve a target-area equation.
Bring the unit skills together
Use these 4 new cases to check your reasoning. Choices are temporary and are not saved to an account, sent to staff or recorded as school grades. Download your current review choices before leaving or reloading.
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Use the skills in a new task
Write a design analysis for a fictional rectangle with perimeter 24. Derive its area rule and positive-length domain. Calculate areas at widths 2,3,4 and their first differences. Use A = 36 − (w − 6)² to prove the maximum. Finally find all widths with area 27 and verify both. No physical construction is needed.
Work on paper or in your own document. Explain the choices you made.
- Derive A(w) = w(12 − w) with 0 < w < 12.
- Calculate areas 20,27,32 and first differences 7 and 5.
- Explain the maximum 36 at width 6 using a nonnegative square.
- Solve the area-27 equation to get widths 3 and 9.
- Verify 3 × 9 = 27 and positive sides for both orientations.
Compare with a model response
Perimeter 24 gives w + L = 12, so A = w(12 − w) and 0 < w < 12. Widths 2,3,4 give 20,27,32; changes are 7 and 5, so the model is not linear. Since A = 36 − (w − 6)², the maximum is 36 at w = 6. Area 27 gives (w − 3)(w − 9) = 0, so widths 3 and 9 both work.
A different response can also meet the criteria. Compare the evidence and reasoning, rather than copying the wording.