Study guide: Build and Use a Quadratic Area Model
Create a constrained area rule, inspect its changes, find a maximum, and solve a target-area equation.
Ideas to explain in your own words
- Connect a quadratic expression with a table and a constrained maximum.
- Translate a fixed perimeter into a quadratic rule and domain.
- Distinguish nonconstant first differences from a linear rule.
- Solve a quadratic area equation and retain both valid widths.
Watch your reasoning
- A(3) = 3(10 − 3) = 3 × 7 = 21. In the other form, 25 − (3 − 5)² = 25 − 4 = 21, providing an independent check.
- Dividing 2(w + L) = 28 by two gives w + L = 14. Thus L = 14 − w and A = w(14 − w), with 0 < w < 14 for positive sides.
- Subtract in order: 20 − 11 = 9, 27 − 20 = 7, and 32 − 27 = 5. The −2 values are second differences, not the requested first differences.
- w² − 10w + 21 = (w − 3)(w − 7), so w = 3 or 7. Both give 3 × 7 = 21 and lie in the positive-length domain.
A worked review example
Find a rectangle’s largest modelled area: A(2) = 2 × 8 = 16; A(5) = 5 × 5 = 25; A(8) = 8 × 2 = 16 square units. The equal areas at 2 and 8 reflect symmetry around w = 5. Write an Area Model from a Perimeter: At w = 3, L = 12 − 3 = 9 and A = 3 × 9 = 27 square units. The perimeter check is 2(3 + 9) = 24 units. Inspect Changing Area Increments: A(2) − A(1) = 16 − 9 = 7, while A(3) − A(2) = 21 − 16 = 5. Their difference is 5 − 7 = −2, so equal width steps do not give equal area steps. Find Widths for a Target Area: Widths 2 and 8 both work: 2(10 − 2) = 16 and 8(10 − 8) = 16. The designs are 2 by 8 and 8 by 2, each with positive sides.
Make a practice plan
- Try the supplied lesson problems without looking at the answers.
- Compare your method with the worked examples and explain any correction.
- Use the new review cases, then complete the transfer task on paper or in your own document.