Chapter 9.1, Problem 83AYU

Chicago ln Chicago, the road system is set up like a Cartesian plane, where streets are indicated by the number of inlines they are from Madison Street and State Street. For example, Wrigley Field in Chicago is located at 1060 West Addison, which is 10 inlines west of State Street and 36 inlines 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 $x\text{-axis}$.

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 35th and Princeton, which is 3 inlines west of State Street and 35 inlines 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.

To determine

To find:

a. Write the location of Wrigley Field using rectangular coordinates.

Answer to Problem 83AYU

Solution:

a. $\left(-10, 36\right)$

Explanation of Solution

Given:

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 $x\text{-axis}$.

Calculation:

a. The location of Wrigley Field using Cartesian coordinates.

As Wrigley city is 10 blocks left in west direction, thus its $x\text{-coordinate}$ represented by $-10$ and its 36 units in north direction along positive $y\text{-axis}$, thus its $y\text{-coordinate}$ represented by 36.

$\left(x, y\right)=\left(-10, 36\right)$

To determine

To find:

b. Write the location of Wrigley Field using polar coordinates. Use the east direction for the polar axis. Express $\theta$ in degrees.

Answer to Problem 83AYU

Solution:

b.

Explanation of Solution

Given:

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 $x\text{-axis}$.

Calculation:

b. The location of Wrigley Field using polar coordinates

$\left(r, \theta \right)=\left(\sqrt{x^2+y^2}, {\tan }^{-1}\left(\frac{y}{x}\right)\right)$

$=\left(\sqrt{1396}, {\tan }^{-1}\left(\frac{36}{-10}\right)\right)$

$=\left(37.36, \left(180-74.5\right)°\right)$

$=\left(37.36,105.5°\right)$

To determine

To find:

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.

Answer to Problem 83AYU

Solution:

c. $\left(-3, -35\right)$

Explanation of Solution

Given:

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 $x\text{-axis}$.

Calculation:

c. U.S. Cellular Field is 3 blocks left in west direction, thus its $x\text{-coordinate}$ represented by $-3$ and its 35 units in south direction along negative $y\text{-axis}$, thus its $y\text{-coordinate}$ represented by $-35$.

$\left(x, y\right)=\left(-3, -35\right)$

To determine

To find:

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

Answer to Problem 83AYU

Solution:

d. $\left(35.13, 265.1°\right)$

Explanation of Solution

Given:

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 $x\text{-axis}$.

Calculation:

d. The location of Wrigley Field using polar coordinates

$\left(r, \theta \right)=\left(\sqrt{x^2+y^2}, {\tan }^{-1}\left(\frac{y}{x}\right)\right)$

$=\left(\sqrt{1234}, {\tan }^{-1}\left(\frac{-35}{-3}\right)\right)$

$=\left(35.13, \left(180+85.1\right)°\right)$

$=\left(35.13,265.1°\right)$