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How can you maximize your happiness under a budget? When does a function reach its minimum value? When does a curve change direction? The calculus of extrema explains these "extreme" situations.

In the above diagram, \(P\) 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 \(T?\)

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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.

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A cylinder made of an iron plate can contain \(3000 \text{ cm}^3\) 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.)

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

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

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