Practice Geometric Optimization in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
Finding the best geometric configuration — the shape that maximizes area, minimizes perimeter, uses the least material, or achieves some other optimal outcome — subject to given constraints.
What rectangle with fixed perimeter has the most area? A square!
Example 1
mediumA farmer has P = 60 m of fence to enclose a rectangular paddock against a straight wall (so only 3 sides need fencing). Find the dimensions that maximise the area.
Example 2
easyA rectangle has perimeter P = 40 cm. Using the formula maximum area = P^2/16, compute the maximum area and the dimensions of the optimal rectangle.
Example 3
easyTwo rectangles have the same perimeter of 24 cm: one is 8 \times 4 cm and one is 6 \times 6 cm. Which has greater area? Does this match the P^2/16 maximum?
Example 4
hardA rectangular box with a square base and no lid must have volume V = 32 cm^3. Find the dimensions that minimise the total surface area (base + 4 sides).