Question

In Chicago, the road system is set up like a Cartesian plane, where streets are indicated by the number of blocks they are from Madison Street and State Street. For example, Wrigley Field in Chicago is located at 1060 West Addison, which is 10 blocks west of State Street and 36 blocks north of Madison Street. Treat the intersection of Madison Street and State Street as the origin of a coordinate system, with east being the positive \(x\) -axis. (a) Write the location of Wrigley Field using rectangular coordinates. (b) Write the location of Wrigley Field using polar coordinates. Use the east direction for the polar axis. Express \(\theta\) in degrees. (c) Guaranteed Rate Field, home of the White \(\operatorname{Sox},\) is located at 35 th and Princeton, which is 3 blocks west of State Street and 35 blocks south of Madison. Write the location of Guaranteed Rate Field using rectangular coordinates. (d) Write the location of Guaranteed Rate Field using polar coordinates. Use the east direction for the polar axis. Express \(\theta\) in degrees.

Short answer

Wrigley Field: Rectangular (-10, 36), Polar (36.4, 106\(^{\circ}\)). Guaranteed Rate Field: Rectangular (-3, -35), Polar (35.1, 264.9\(^{\circ}\)).

Step-by-step solution

- Determine rectangular coordinates of Wrigley Field

Given Wrigley Field is 10 blocks west of State Street and 36 blocks north of Madison Street. This means it is located at (-10, 36) on the Cartesian coordinate system. The negative value indicates it is west of State Street, and the positive value indicates it is north of Madison Street.

- Convert rectangular coordinates of Wrigley Field to polar coordinates

To convert from rectangular coordinates \((x, y)\) to polar coordinates \((r, \theta)\), use the following formulas: \[ r = \sqrt{x^2 + y^2} \] and \[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \] For Wrigley Field, \(x = -10, y = 36\). Calculate\( r\): \[ r = \sqrt{(-10)^2 + 36^2} = \sqrt{100 + 1296} = 36.4 \] Now, calculate \(\theta\) in degrees, noting that the angle will be in the second quadrant: \[ \theta = \tan^{-1}\left(\frac{36}{-10}\right) = \tan^{-1}(-3.6) \approx -74^{\circ} \] Because it's in the second quadrant, \(\theta = 180^{\circ} - 74^{\circ} = 106^{\circ} \) Therefore, the polar coordinates are \( (36.4, 106^{\circ}) \)

- Determine rectangular coordinates of Guaranteed Rate Field

Given Guaranteed Rate Field is 3 blocks west of State Street and 35 blocks south of Madison Street. This means it is located at (-3, -35) on the Cartesian coordinate system. The negative \(x\)-value indicates west of State Street, and the negative \(y\)-value indicates south of Madison Street.

- Convert rectangular coordinates of Guaranteed Rate Field to polar coordinates

To convert from rectangular coordinates \((x, y)\) to polar coordinates \((r, \theta)\), use these formulas: \] r = \sqrt{x^2 + y^2} \[ and \[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \] For Guaranteed Rate Field, \(x = -3, y = -35\). Calculate \( r\): \[ r = \sqrt{(-3)^2 + (-35)^2} = \sqrt{9 + 1225} = 35.1 \] Now, calculate \( \theta \) in degrees, noting that the angle will be in the third quadrant: \[ \theta = \tan^{-1}\left(\frac{-35}{-3}\right) = \tan^{-1}(11.67) \approx 84.9^{\circ} \] Since it's in the third quadrant, \(\theta = 180^{\circ} + 84.9^{\circ} = 264.9^{\circ} \) Thus, the polar coordinates are \( (35.1, 264.9^{\circ}) \)

Key concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, represent points in a plane using two values: the x-coordinate and the y-coordinate. This system is based on a grid made up of two perpendicular lines, the x-axis (horizontal) and the y-axis (vertical). These two values indicate how far a point is from the origin (0,0), where the axes intersect. For instance, in the exercise, Wrigley Field being 10 blocks west of State Street and 36 blocks north of Madison Street translates to the coordinates (-10, 36).

The negative x-value (-10) reflects the westward direction, and the positive y-value (36) indicates the northward direction. It's a straightforward way to describe positions in a mapped layout of a city or any surveyed area.

Key points:

• X-axis indicates horizontal distance
• Y-axis indicates vertical distance
• Negative values show directions opposite to the positive axis

Polar Coordinates

Polar coordinates offer an alternative way to define the location of a point. Instead of using a grid of perpendicular lines, polar coordinates use a distance from a reference point and an angle from a reference direction (usually the positive x-axis). These coordinates are given as a pair \(r, \theta\), where \(r\) is the radius or distance from the origin, and \(\theta\) is the angle.

To convert from rectangular coordinates \(x, y\) to polar coordinates \(r, \theta\), we use:

• \[ r = \sqrt{x^2 + y^2} \]
• \[ \theta = \tan^{-1}(y/x) \]

For Wrigley Field with coordinates \(-10, 36\), you first find the radius: \[ \r = \sqrt{(-10)^2 + 36^2} = 36.4 \]

Next, you find the angle. Since it's in the second quadrant: \[ \theta =\tan^{-1}(36/-10) = -74^{\circ}, \] then corrected to 106°. Thus, the polar coordinates are (36.4, 106°). This method provides a clear way to understand positions through angles and distances, especially helpful in navigation and trigonometry.

Cartesian Plane

The Cartesian plane, named after René Descartes, is a two-dimensional plane defined by a horizontal line (x-axis) and a vertical line (y-axis). These axes divide the plane into four quadrants. Each point on the plane is determined by an ordered pair \(x, y\), where \(x\) tells the horizontal position and \(y\) tells the vertical position.

The point (0,0) is called the origin. In this layout, positive values of x lie to the right of the origin, and negative values lie to the left. Similarly, positive y-values extend upward from the origin, while negative y-values extend downward.

For instance, Guaranteed Rate Field being 3 blocks west and 35 blocks south translates to the coordinates \(-3, -35\). Key points of the Cartesian Plane:

• Origin at (0,0)
• Four quadrants:
• Top right: positive x, positive y
• Top left: negative x, positive y
• Bottom left: negative x, negative y
• Bottom right: positive x, negative y

This system is widely used for plotting graphs and solving geometrical problems.