Question

1. a ranger in a lookout tower spots two fires on the campground below. fire a is 75 meters east and 40 meters south of the tower. fire b is 37 meters west and 64 meters south of the tower. if there is a fire hydrant located three - fourths of the way from fire a to fire b, find the location of the fire hydrant relative to the tower.

Explanation:

Step1: Set up a coordinate - system

Let the tower be at the origin \((0,0)\). Fire A is at the point \((75, - 40)\) (75 meters east and 40 meters south of the tower). Fire B is at the point \((75 - 37,-40 - 64)=(38,-104)\) (37 meters west and 64 meters south of Fire A).

Step2: Use the section - formula

The section - formula for a point \(P(x,y)\) that divides the line segment joining \(A(x_1,y_1)\) and \(B(x_2,y_2)\) in the ratio \(m:n\) is \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here, \(m = 3\), \(n = 1\), \(x_1=75\), \(y_1=-40\), \(x_2 = 38\), \(y_2=-104\).
\[x=\frac{3\times38 + 1\times75}{3 + 1}=\frac{114+75}{4}=\frac{189}{4}=47.25\]
\[y=\frac{3\times(-104)+1\times(-40)}{3 + 1}=\frac{-312-40}{4}=\frac{-352}{4}=-88\]

Answer:

The location of the fire hydrant is \(47.25\) meters east and \(88\) meters south of the tower.