Question

a large garden has a set of underground sprinklers. suppose the center of the garden is located at the origin of a grid with intervals measured in feet. one sprinkler is positioned at (2,4) and reaches a radius of 8 feet. the second sprinkler is positioned at (-6,-4) and reaches a radius of 6 feet. the system of inequalities represents this scenario.
$$\begin{cases}(x - 2)^2+(y - 4)^2leq64\\(x + 6)^2+(y + 4)^2leq36end{cases}$$
which location is in the region for plants receiving the maximum amount of water?
(-3,-1)
(-7,1)
(1,-5)
(2,1)

Explanation:

Step1: Substitute \((-3,-1)\) into the first inequality

$$(x - 2)^2+(y - 4)^2=(-3 - 2)^2+(-1 - 4)^2=25 + 25=50\leq64$$
$$(x + 6)^2+(y + 4)^2=(-3+6)^2+(-1 + 4)^2=9+9 = 18\leq36$$

Step2: Substitute \((-7,1)\) into the first inequality

$$(x - 2)^2+(y - 4)^2=(-7 - 2)^2+(1 - 4)^2=81+9 = 90>64$$

Step3: Substitute \((1,-5)\) into the second inequality

$$(x + 6)^2+(y + 4)^2=(1 + 6)^2+(-5 + 4)^2=49+1=50>36$$

Step4: Substitute \((2,1)\) into the second inequality

$$(x + 6)^2+(y + 4)^2=(2 + 6)^2+(1 + 4)^2=64 + 25=89>36$$

Answer:

\((-3,-1)\)