Calculus
# Extrema

$P$ is on the arc $AB$ of a quarter of a circle $OAB$ with radius $r=13,$ and the line segment $\overline{PQ}$ is perpendicular to $\overline{OA}.$ If $\square PQRS$ is a square and the area of the shaded region is $T,$ what is the length of $\overline{PQ}$ that maximizes $T?$

In the above diagram,A shop sells $500$ smartphones a week for $\$450$ each. A market survey shows that each decrease of $\$5$ on the price will result in the sale of an additional $10$ 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.

$3000 \text{ cm}^3$ 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.)

A closed cylinder made of an iron plate can contain$45\text{ cm},$ what is the maximum volume of the cylinder?

If the sum of the radius and height of a cylinder is$r=60\text{ cm}.$ What is the radius of the cylinder (in cm) that maximizes the volume of the cylinder?

As shown in the diagram above, a cylinder is inscribed in a right circular cone with base radius