What you’ll learn
Solve geometric design optimization problems.
Key idea
For a fixed side sum, completing the square of the area expression identifies the maximum at equal sides.
Respect positive dimensions and use the balanced square equality case; no three-sided fence assumption.
Worked example
Enter the exact value or ordered values in the format requested below.
A rectangle uses exactly 68 m of fencing on all four sides. Find its greatest possible area in square meters.
- Positive sides x and 34-x give A=x(34-x)=289-(x-17)^2.
- The nonnegative square term is minimized at x=17, giving maximum 289 m².
Answer: 289
Quick Check
Evaluate a worked attempt at solve geometric design optimization problems, identify the first invalid step, and provide the correction.
A four-sided rectangular garden has fencing 16 m. Step 1 gets width+length=8. Step 2 chooses sides 1 and 7, claiming these maximize area 7. Locate and repair the first invalid step.