What is the concept of a control volume in fluid mechanics, and how might rectangular areas be used to define its boundaries?
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- What are measurable sets and how is their area (or measure) defined?
- How would you find the center of mass of a uniform rectangular plate?
- What is Gaussian curvature and how does it affect the area of shapes on a surface?
- Consider a rectangle with sides of integer lengths and an area of n. What are the possible perimeters of such rectangles?
- Consider a rectangle whose diagonals intersect at a point $(h, k)$ and whose sides are parallel to the coordinate axes. If the length of the diagonal is $2d$, what can you say about the possible areas of the rectangle?
- A rectangular park has a length that is twice its width. If the width is 50 meters, what is the area of the park?
- How is the parallel axis theorem used to find the moment of inertia of a rectangle about an axis not passing through its centroid?
- How can the concept of area be used to derive the formula for the volume of a rectangular prism?
- A map has a scale of 1 1000. A rectangular park on the map measures 10 cm by 5 cm. What is the actual area of the park in square meters?
- What is the area of a rectangle formed by the vectors $\vec{u} = (a, b)$ and $\vec{v} = (c, d)$ in the plane (consider the parallelogram formed by these vectors)?
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