Question

a pair of fire towers uses technology to search for forest fires. let the center of the forest be the origin on a coordinate grid. the first fire tower is located 5 km east and 8 km south of the center, and the second fire tower is located 7 km west and 4 km north of the center. the technology at the first tower can locate fires within a radius of 10 km and the second tower within a radius of 9 km. the system of inequalities represents this scenario.
\\(\

$$\begin{cases}(x - 5)^2+(y + 8)^2\\leq100\\\\(x + 7)^2+(y - 4)^2\\leq81\\end{cases}$$

\\)
at which location, from the origin, would a fire be detected by both towers?
2 km east and 3 km south
3 km east and 2 km north
2 km west and 1 km north
1 km west and 2 km south

Explanation:

Step1: Check option A (2 km east and 3 km south, so \(x = 2\), \(y=-3\))

For the first inequality \((x - 5)^2+(y + 8)^2\):
Substitute \(x = 2\) and \(y=-3\) into \((x - 5)^2+(y + 8)^2=(2 - 5)^2+(-3 + 8)^2=(-3)^2+5^2=9 + 25=34\leq100\).
For the second - inequality \((x + 7)^2+(y - 4)^2\):
Substitute \(x = 2\) and \(y=-3\) into \((x + 7)^2+(y - 4)^2=(2+7)^2+(-3 - 4)^2=9^2+(-7)^2=81 + 49=130>81\). So option A is not correct.

Step2: Check option B (3 km east and 2 km north, so \(x = 3\), \(y = 2\))

For the first inequality \((x - 5)^2+(y + 8)^2\):
Substitute \(x = 3\) and \(y = 2\) into \((x - 5)^2+(y + 8)^2=(3 - 5)^2+(2 + 8)^2=(-2)^2+10^2=4 + 100=104>100\). So option B is not correct.

Step3: Check option C (2 km west and 1 km north, so \(x=-2\), \(y = 1\))

For the first inequality \((x - 5)^2+(y + 8)^2\):
Substitute \(x=-2\) and \(y = 1\) into \((x - 5)^2+(y + 8)^2=(-2 - 5)^2+(1 + 8)^2=(-7)^2+9^2=49+81 = 130>100\). So option C is not correct.

Step4: Check option D (1 km west and 2 km south, so \(x=-1\), \(y=-2\))

For the first inequality \((x - 5)^2+(y + 8)^2\):
Substitute \(x=-1\) and \(y=-2\) into \((x - 5)^2+(y + 8)^2=(-1 - 5)^2+(-2 + 8)^2=(-6)^2+6^2=36 + 36=72\leq100\).
For the second inequality \((x + 7)^2+(y - 4)^2\):
Substitute \(x=-1\) and \(y=-2\) into \((x + 7)^2+(y - 4)^2=(-1 + 7)^2+(-2 - 4)^2=6^2+(-6)^2=36+36 = 72\leq81\).

Answer:

D. 1 km west and 2 km south