Question Statement

A cell phone company has installed three signal towers $A,B$ and $C$ at different locations to provide service to their customers. Tower $A$ is located at the position $(-2,10)$ and covers an area up to 8 km, tower $B$ is located at $(-3,5)$ and covers an area up to 6 km and tower $C$ is located at $(10,-1)$ and covers an area up to 5 km. A man is standing at a position of $(12,3)$. Which tower will provide service to the man?

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Background and Explanation

This problem involves the geometry of circles in the coordinate plane. Each tower covers a circular area, and we need to determine if the man's position falls within any of these circles. A point $(x_1,y_1)$ lies inside a circle if the distance from the point to the center is less than the radius.

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Solution

Each tower covers a circular region. We write the equation of each circle and substitute the man's coordinates $(12,3)$ to check if he lies within the coverage area. For a circle with center $(h,k)$ and radius $r$, the equation is $(x-h{)}^2+(y-k{)}^2=r^2$. If substituting a point gives a value less than $r^2$, the point is inside; if greater than $r^2$, it is outside.

Setting Up the Circle Equations

Tower A: Center $(-2,10)$, radius $8$ $[x-(-2){]}^2+(y-10{)}^2=8^2$ $(x+2{)}^2+(y-10{)}^2=64$ or $(x+2{)}^2+(y-10{)}^2-64=0$

Tower B: Center $(-3,5)$, radius $6$ $(x-(-3){)}^2+(y-5{)}^2=6^2$ $(x+3{)}^2+(y-5{)}^2=36$ or $(x+3{)}^2+(y-5{)}^2-36=0$

Tower C: Center $(10,-1)$, radius $5$ $(x-10{)}^2+(y-(-1){)}^2=5^2$ $(x-10{)}^2+(y+1{)}^2=25$ or $(x-10{)}^2+(y+1{)}^2-25=0$

Checking the Position of Point $(12,3)$

For Tower A:

Substitute $(x_1,y_1)=(12,3)$ into equation (i): $(x_1+2{)}^2+(y_1-10{)}^2-64=(12+2{)}^2+(3-10{)}^2-64$ $=(14{)}^2+(-7{)}^2-64$ $=196+49-64$ $=181>0$

Since the result is positive, the point lies outside the circle. Thus, the man is not in the coverage area of tower $A$.

For Tower B:

Substitute $(x_1,y_1)=(12,3)$ into equation (ii): $(x_1+3{)}^2+(y_1-5{)}^2-36=(12+3{)}^2+(3-5{)}^2-36$ $=(15{)}^2+(-2{)}^2-36$ $=225+4-36$ $=193>0$

Since the result is positive, the point lies outside the circle. Thus, the man is not in the coverage area of tower $B$.

For Tower C:

Substitute $(x_1,y_1)=(12,3)$ into equation (iii): $(x_1-10{)}^2+(y_1+1{)}^2-25=(12-10{)}^2+(3+1{)}^2-25$ $=(2{)}^2+(4{)}^2-25$ $=4+16-25$ $=-5<0$

Since the result is negative, the point lies inside the circle.

Conclusion

Only Tower $C$ yields a negative value when testing the man's position, indicating he is within its 5 km coverage radius.

Therefore, Tower $C$ is providing service to the man.

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Key Formulas or Methods Used

• Standard equation of a circle: $(x-h{)}^2+(y-k{)}^2=r^2$ where $(h,k)$ is the center and $r$ is the radius
• Point-circle position test: For a point $(x_1,y_1)$ and circle $(x-h{)}^2+(y-k{)}^2=r^2$:
  • If $(x_1-h{)}^2+(y_1-k{)}^2-r^2<0$: point lies inside the circle
  • If $(x_1-h{)}^2+(y_1-k{)}^2-r^2>0$: point lies outside the circle
  • If $(x_1-h{)}^2+(y_1-k{)}^2-r^2=0$: point lies on the circle
• Distance formula application: Comparing squared distance from center to squared radius avoids calculating square roots

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Summary of Steps

1. Identify circle parameters: Extract center coordinates $(h,k)$ and radius $r$ for each tower
2. Write circle equations: Form $(x-h{)}^2+(y-k{)}^2=r^2$ for towers $A$, $B$, and $C$
3. Test the point $(12,3)$: Substitute $x=12$ and $y=3$ into each circle's equation (rearranged to equal zero)
4. Evaluate the sign:
   • Tower A: $181>0$ → Outside coverage
   • Tower B: $193>0$ → Outside coverage
   • Tower C: $-5<0$ → Inside coverage
5. Conclusion: Only Tower $C$ covers the man's location at $(12,3)$