Question

A cell tower is located 3 miles east and 4 miles north of the center of a small town. The cell tower has a coverage radius of 3 miles.
Start by drawing a diagram of this situation. Your diagram might include a coordinate plane, the cell tower, and a circle representing the cell tower's coverage boundary.

A house is located 6 miles north of the center of the town and is to the east of the cell tower. If the house lies on the boundary of the cell tower's coverage, how far east of the center of the town is the house?

Answer 1

Answer:

Step-by-step explanation:

1) Find the diagram to the question attached. If the cell tower is located 3 miles east and 4 miles north of the center of a small town, then the coordinate of location will be (x1, y2) = (3,4). The east is along the positive x component (x1= 3) while the north is along the positive y component (y1 = 4).

If the cell tower has a coverage radius of 3 miles, then the radius of the circle will be 3 miles. We can find the equation of the circle according to the equation.

(x-x1)²+(y-y1)² = r² where:

(a,b) is the centre of the circle and r is the radius

Substituting the given coordinate and radius into the formula:

(x-3)²+(y-4)² = 3²

(x-3)²+(y-4)² = 9

2) If a house is located 6 miles north of the center of the tower, this means that y = 6. We can therefore calculate how far east of the center of the town is the house by substituting y = 6 into the resulting equation in 1 and finding x as shown;

(x-3)²+(y-4)² = 9

at y = 6, the equation becomes:

(x-3)²+(6-4)² = 9

(x-3)²+2² = 9

(x-3)² = 9-4

(x-3)² = 5

Take the square root of both sides

√(x-3)² = √5

x-3 = √5

x = √5 + 3

x ≈ 5.24

This means that the house is 5.24miles of east from the center of the town.