Can the Smith family receive the radio signal from Maintown?A tower in Maintown sends radio signal a certain distance (in miles) according to the equation: `x^2+ y^2= 8100` . The Smith family lives...

Can the Smith family receive the radio signal from Maintown?

A tower in Maintown sends radio signal a certain distance (in miles) according to the equation: `x^2+ y^2= 8100` . The Smith family lives 50 miles south and 70 miles east of Maintown. 

Asked on by jpwong96

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vixen999's profile pic

vixen999 | High School Teacher | (Level 2) Adjunct Educator

Posted on

The first thing you should do when looking at this problem is identify that the equation given in the problem is the formula for a circle.  So, the tower is basically in the center of a circle and emits signal in all directions (360 degrees).  Thus, anyone living within (or on) this circle will be able to receive the signal,

The problem does not state whether "x" is the north/south coordinate or the east/west coordinate.  Therefore, we can assume that the x coordinate is the east/west coordinate (right/left on a coordinate plane) and the y coordinate is the north/south coordinate (up/down on a coordinate plane).  If we substitute the values in for "x" and "y," we get 

70^2 + 50^2 = 8100

4900 + 2500 = 8100

7400 = 8100

We can determine from this information that the Smith family does not live on the circle, but lives within the circle.  This means that the Smith family will be able to obtain the signal from the tower.  Anyone living past the "boundary" of the circle would not be able to obtain service.

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lemjay's profile pic

lemjay | High School Teacher | (Level 3) Senior Educator

Posted on

The equation `x^2+y^2 = 1800` is a circle. It's center is (0,0) and radius is 90.

The center(0,0) represents the location of the tower. And the radius indicates that all residences or establishments within 90 miles from the tower can receive radio signals. 

The location of Smith family is 50 miles south and 70 miles east of the tower in Maintown. This can be express in x and y coordinates which is (70, -50). 

The points (0,0) and (70,-50) can be used to determine if the Smith family can receive radio signals from the Maintown. This can be done using the distance formula between two points. If the distance is less than or equal to 90 miles, then the Smith family can receive signals.

`d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2) = sqrt[(70-0)^2+(-50-0)^2] = sqrt7400 = 86.02`

So the Smith family is 86.02 miles away from the tower and they are still within the 90 mile radius. Hence, the Smith family can receive signals.

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