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 (maximum area, minimum distance, etc.).
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).