Question

the town of lakehorn is built on a grid system. town hall is located downtown at point (0,0). a new school is located 3 miles north and 2 miles east of town hall. only students who live outside a 5 - mile radius from the school are eligible to ride the school bus. which of the following students are eligible to ride the bus? select all that apply. charity lives 1 mile due south of town hall. thomas lives 3 miles south and 4 miles east of the school. kaleb lives 2 miles south and 4 miles east of the town hall. marinna lives 2 miles north and 3 miles west of the town hall.

Explanation:

Step1: Find the distance formula

The distance $d$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. The school is at the point $(2,3)$ (2 miles east and 3 miles north of the origin which is the town - hall) and we will find the distance of each student's location from the school.

Step2: Calculate Marina's distance

Marina's location relative to the town - hall is $(- 3,2)$ (3 miles west and 2 miles north). Using the distance formula with $(x_1 = 2,y_1 = 3)$ (school's coordinates) and $(x_2=-3,y_2 = 2)$:
\[

$$\begin{align*} d_{Marina}&=\sqrt{(-3 - 2)^2+(2 - 3)^2}\\ &=\sqrt{(-5)^2+(-1)^2}\\ &=\sqrt{25 + 1}\\ &=\sqrt{26}\approx5.1\text{ miles} \end{align*}$$

\]

Step3: Calculate Kaleb's distance

Kaleb's location relative to the town - hall is $(4,-2)$ (4 miles east and 2 miles south). Using the distance formula with $(x_1 = 2,y_1 = 3)$ and $(x_2 = 4,y_2=-2)$:
\[

$$\begin{align*} d_{Kaleb}&=\sqrt{(4 - 2)^2+(-2 - 3)^2}\\ &=\sqrt{(2)^2+(-5)^2}\\ &=\sqrt{4 + 25}\\ &=\sqrt{29}\approx5.4\text{ miles} \end{align*}$$

\]

Step4: Calculate Thomas's distance

Thomas's location relative to the town - hall is $(4,-3)$ (4 miles east and 3 miles south). Using the distance formula with $(x_1 = 2,y_1 = 3)$ and $(x_2 = 4,y_2=-3)$:
\[

$$\begin{align*} d_{Thomas}&=\sqrt{(4 - 2)^2+(-3 - 3)^2}\\ &=\sqrt{(2)^2+(-6)^2}\\ &=\sqrt{4+36}\\ &=\sqrt{40}\approx6.3\text{ miles} \end{align*}$$

\]

Step5: Calculate Charity's distance

Charity's location relative to the town - hall is $(0,-1)$ (0 miles east - west and 1 mile south). Using the distance formula with $(x_1 = 2,y_1 = 3)$ and $(x_2 = 0,y_2=-1)$:
\[

$$\begin{align*} d_{Charity}&=\sqrt{(0 - 2)^2+(-1 - 3)^2}\\ &=\sqrt{(-2)^2+(-4)^2}\\ &=\sqrt{4 + 16}\\ &=\sqrt{20}\approx4.5\text{ miles} \end{align*}$$

\]
Since students outside a 5 - mile radius from the school are eligible to ride the bus, Marina, Kaleb, and Thomas have distances greater than 5 miles.

Answer:

Marina, Kaleb, Thomas