Questions: A map of a wilderness area is drawn with the origin placed at the parking area. Two fire observation platforms are located at points A and B. If a fire is located at point C, determine the distance to the fire from each observation platform. Which observation tower is closer to the fire? Assume that the distance between grid lines is 1 km. Round your answers to one decimal place, if necessary.

A map of a wilderness area is drawn with the origin placed at the parking area. Two fire observation platforms are located at points A and B. If a fire is located at point C, determine the distance to the fire from each observation platform. Which observation tower is closer to the fire? Assume that the distance between grid lines is 1 km. Round your answers to one decimal place, if necessary.

Transcript text: A map of a wilderness area is drawn with the origin placed at the parking area. Two fire observation platforms are located at points $A$ and $B$. If a fire is located at point $C$, determine the distance to the fire from each observation platform. Which observation tower is closer to the fire? Assume that the distance between grid lines is 1 km . Round your answers to one decimal place, if necessary.

Solution

Solution Steps

Step 1: Identify the coordinates

Point A: (-3, 7)
Point C (fire location): (6, 5)

Step 2: Use the distance formula

The distance formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Step 3: Calculate the distance from A to C

Substitute the coordinates of A and C into the distance formula: \[ d_{AC} = \sqrt{(6 - (-3))^2 + (5 - 7)^2} \] \[ d_{AC} = \sqrt{(6 + 3)^2 + (5 - 7)^2} \] \[ d_{AC} = \sqrt{9^2 + (-2)^2} \] \[ d_{AC} = \sqrt{81 + 4} \] \[ d_{AC} = \sqrt{85} \] \[ d_{AC} \approx 9.2 \, \text{km} \]