Chicago 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) U.S. Cellular Field, home of the White 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 U.S. Cellular Field using rectangular coordinates.

(d) Write the location of U.S. Cellular Field using polar coordinates. Use the east direction for the polar axis. Express $\theta$in degrees.

The Wrigley Field is $10$blocks west of State Street and $36$blocks north of Madison Street.

And the origin is the intersection of Madison Street and State Street.

Using the reference line for positive $x$axis as East direction.

So the location of Wrigley Field using rectangular coordinates can be written as$\left(-10,36\right)$where the origin is the intersection of Madison Street and State Street.

The rectangular coordinates of Wrigley Field is $\left(-10,36\right)$. So

$$\begin{gathered} x=-10 \\ y=36 \end{gathered}$$

Now the relation between $x,y,r$is given as $r^2=x^2+y^2$. So the value of $r$is given as

$$\begin{gathered} r^2=x^2+y^2 \\ {r}^2={(-10)}^2+{\left(36\right)}^2{r}^2=100+1296 \\ {r}^2=1396 \\ r=\sqrt{1396} \\ r\approx 37.36 \end{gathered}$$

The angle $\theta$can be found by using the relation $\theta ={\tan }^{-1}\left(\frac{y}{x}\right)$. Now as the point $\left(-10,36\right)$is in the second quadrant so we add $180°$to the angle to get the angle as

$$\begin{gathered} \theta =180°+{\tan }^{-1}\left(\frac{36}{-10}\right) \\ \theta =180°+{\tan }^{-1}(-3.6) \\ \theta =180°+(-74.48°) \\ \theta =105.52° \end{gathered}$$

So the location of Wrigley Field using polar coordinates is $(r,\theta )=(37.36,105.52°)$.

U.S. Cellular Field is $3$blocks west of State Street and $35$blocks south of Madison.

So it can be written in form rectangular coordinate as $\left(-3,-35\right)$where the origin is the intersection of Madison Street and State Street.

The rectangular coordinates of U.S. Cellular Field is $\left(-3,-35\right)$. So

$$\begin{gathered} x=-3 \\ y=-35 \end{gathered}$$

Now the relation between $x,y,r$is given as $r^2=x^2+y^2$. So the value of $r$is given as

$$\begin{gathered} r^2=x^2+y^2 \\ {r}^2={(-3)}^2+{(-35)}^2 \\ {r}^2=9+1225 \\ {r}^2=1234 \\ r=\sqrt{1234} \\ r\approx 35.13 \end{gathered}$$

The angle$\theta$can be found by using the relation $\theta ={\tan }^{-1}\left(\frac{y}{x}\right)$. Now as the point $\left(-3,-35\right)$is in the third quadrant so we add $180°$to the angle to get the angle as

$$\begin{gathered} \theta =180°+{\tan }^{-1}\left(\frac{-35}{-3}\right) \\ \theta =180°+{\tan }^{-1}\left(\frac{35}{3}\right) \\ \theta \approx 180°+85.1° \\ \theta =265.1° \end{gathered}$$

So the location of U.S. Cellular Field using polar coordinates is $(r,\theta )=(35.13,265.1°)$.