1. Find the dimensions of a rectangle with a fixed perimeter of 20 units that maximizes the area.
HANDWRITTEN THEN BOX THE FINAL ANSWERS. THESE ARE OPTIMIZATION PROBLEMS USING EXTREMA
1. Find the dimensions of a rectangle with a fixed perimeter of 20 units that maximizes the area.
2. Find the dimensions of a cylindrical container with a fixed volume of 100 cubic units that minimizes the cost of manufacturing. The cost is directly proportional to the sum of the surface areas of the cylinder's top and bottom, as well as its lateral surface area.
3. A company manufactures and sells a product with a production cost function C(x) = 500x + 10,000 and a revenue function R(x) = 30x, where x represents the production level. Find the production level that maximizes the profit.
4. A car travels a distance of 200 km at a constant speed. Find the speed that minimizes the travel time.
5. A manufacturing process has a fixed cost function C(x) = 2000 + 100x and a production rate function P(x) = 1000 - 10x, where x represents the production rate. Find the production rate that maximizes the efficiency.
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