Question

the range of a wireless speaker, in meters, is represented by the equation $(x - 2)^2+(y - 4)^2 = 100$. which location will be out of the speakers range? (a) $(-8,4)$ (b) $(2,14)$ (c) $(4,16)$ (d) $(8,8)$

Explanation:

Step1: Recall the equation of a circle

The standard form of a circle's equation is \((x - h)^2+(y - k)^2=r^2\), where \((h,k)\) is the center and \(r\) is the radius. For the equation \((x - 2)^2+(y - 4)^2=100\), the center is \((2,4)\) and the radius \(r=\sqrt{100} = 10\).

Step2: Calculate the distance from the center to each point

Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).

• For point \((-8,4)\):

\(d=\sqrt{(-8 - 2)^2+(4 - 4)^2}=\sqrt{(-10)^2+0^2}=\sqrt{100}=10\)

• For point \((2,14)\):

\(d=\sqrt{(2 - 2)^2+(14 - 4)^2}=\sqrt{0^2 + 10^2}=10\)

• For point \((4,16)\):

\(d=\sqrt{(4 - 2)^2+(16 - 4)^2}=\sqrt{2^2+12^2}=\sqrt{4 + 144}=\sqrt{148}\approx12.16>10\)

• For point \((8,8)\):

\(d=\sqrt{(8 - 2)^2+(8 - 4)^2}=\sqrt{6^2+4^2}=\sqrt{36 + 16}=\sqrt{52}\approx7.21<10\)

Answer:

\((4,16)\)