Optimization Problems


In the above diagram, PP is on the arc ABAB of a quarter of a circle OABOAB with radius r=13,r=13, and the line segment PQ\overline{PQ} is perpendicular to OA.\overline{OA}. If PQRS\square PQRS is a square and the area of the shaded region is T,T, what is the length of PQ\overline{PQ} that maximizes T?T?

A shop sells 500500 smartphones a week for $450\$450 each. A market survey shows that each decrease of $5\$5 on the price will result in the sale of an additional 1010 smartphones per week. What price of the smartphone would result in maximum revenue?

Details and Assumptions:

  • The revenue is defined as the product of the number of items sold and the price of each item.

A closed cylinder made of an iron plate can contain 3000 cm33000 \text{ cm}^3 of liquid. What is the radius of the cylinder (in cm) that minimizes the use of the iron plate? (Suppose the thickness of the iron plate is negligible.)

If the sum of the radius and height of a cylinder is 45 cm,45\text{ cm}, what is the maximum volume of the cylinder?

As shown in the diagram above, a cylinder is inscribed in a right circular cone with base radius r=60 cm.r=60\text{ cm}. What is the radius of the cylinder (in cm) that maximizes the volume of the cylinder?


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