91Ó°ÊÓ

Coordinate geometry is a branch of mathematics where geometric problems are solved using coordinates on a plane. It's like taking pictures and labeling them on a map. Instead of saying, "left or right," you can talk about specific spots using numbers.

When we talk about "west" and "south" in coordinate geometry, these directions can translate into negative values on the coordinate plane. For the problem about the FM radio station, we set the radio tower at the origin of our plane, marking it as (0,0).

Then, the home situated 68 km west and 58 km south can be described with the coordinates - -68 km for the west direction (since west is typically negative) - -58 km for the south direction (again, negative because we're moving downward from the origin).

Placing these on a graph helps us visualize exactly where the home is relative to the radio tower, making it much easier to calculate distances.

Radio signal coverage refers to how far a radio signal can travel from its source—often creating a circular coverage area around the transmission point on a flat plane. Think of it like ripples on a pond when you throw a stone. The ripples spread out evenly in all directions until they start to fade.

For the FM radio tower, this means the signal extends as a circle with a radius of 85 km from the tower. This circular reach determines who can tune in to the station without trouble. If you're further out, like at 89.42 km as in the example, you sadly might get static instead of tunes!

Having this mental picture helps everyone understand why we measure distances so precisely when dealing with signals. It's essential for broadcasters and listeners figuring out who gets the clear signal and who doesn't.

Distance calculation is a way to find out how far apart two points are on a coordinate plane. It’s like using a high-tech ruler to measure the exact space between two locations.

To determine if the signal reaches the home, the distance formula helps us check. It's given by:

\[d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\]

Plugging in our numbers, it becomes:- \[d = \sqrt{(-68-0)^2 + (-58-0)^2}\]- After calculating, this results in approximately 89.42 km.

If this distance is longer than the radio's coverage radius (85 km), like in the exercise, unfortunately, the FM waves won't reach the home clearly. Essentially, this difference in distance helps decide whether you'll play music effortlessly or have trouble finding a signal. Everyone using coordinate geometry can quickly make these judgments and ensure their plans meet their needs.