91Ó°ÊÓ

Rectangular coordinates are a way to describe a location on a cartesian plane using two numbers: the x-coordinate and the y-coordinate. For any point on the plane, the x-coordinate tells us how far to move left or right from the origin, and the y-coordinate tells us how far to move up or down. This system is useful because it can easily describe regions, like the pizza slice in our exercise, using inequalities.

In this exercise, the pizza slice is positioned in the first quadrant, which means both x and y are non-negative. The circle slice can be described by the equation \(x^2 + y^2 \leq 1\), representing the area within a circle of radius 1 centered at the origin. Since the slice is in the first quadrant, additional inequalities \(0 \leq x \leq 1\) and \(0 \leq y \leq x\) also describe the constraints for the region corresponding to the pizza slice.

• Origin as the Center: Center is (0, 0).
• Inequalities: \(x \geq 0\) and \(y \leq x\).
• Equation: \(x^2 + y^2 \leq 1\).

Radians are a way of measuring angles based on the radius of a circle. Unlike degrees, which divide a circle into 360 parts, radians divide the circle into units where the angle is equal to the length of the arc on the circle's circumference divided by the radius. Thus, a full circle is \(2\pi\) radians.

In the context of the exercise, the angle covered by the pizza slice is \(\frac{\pi}{4}\) radians. This angle represents an eighth of a full circle, as dividing \(2\pi\) by 8 yields \(\frac{\pi}{4}\). Understanding radians is crucial for converting between polar and rectangular coordinates and defining positions or areas in a circular region.

• Full Circle: \(2\pi\) radians.
• Pizza Slice Angle: \(\frac{\pi}{4}\) radians.
• Radians to Sectors: Useful for describing circular regions.

The first quadrant of the Cartesian coordinate system is where both the x-coordinates and y-coordinates are positive or zero. It is one of the four sections of the coordinate plane. It is important for our problem because the pizza slice lies entirely within this quadrant.

Since the pizza slice is in the first quadrant:

• The x-axis serves as one boundary, meaning all y values are equal to or greater than zero.
• Similarly, the y-axis serves as the other boundary, meaning all x values are equal to or greater than zero.
• These conditions are mathematically expressed with the inequalities \(x \geq 0\) and \(y \geq 0\), and more specifically for this slice, \(y \leq x\).

Understanding these properties helps define the scope of the pizza slice on the coordinate plane.

A circle sector is a portion of a circle, shaped like a "slice" or wedge, enclosed by two radii and an arc. It can be thought of like a slice of a pie. When describing areas in polar coordinates, you use the radius and angle to define the sector.

For our exercise, the circle sector represents one-eighth of a full circle. The circle has a radius of 1 foot, and the arc corresponding to \(\frac{\pi}{4}\) radians, considering a full circle is \(2\pi\) radians. This makes polar coordinates a feasible approach to describing such areas.

• Central Angle of Sector: \(\frac{\pi}{4}\) radians.
• Total Circle Radius: 1 foot.
• Sector Boundaries: Defined by radius and angle.

Understanding the nature of circle sectors aids in solving problems involving circular regions and converting between coordinate systems.