Transmitter problem

Question

A small radio transmitter broadcasts in a 54 mile radius. If you drive along a straight line from a city 73 miles north of the transmitter to a second city 60 miles east of the transmitter, during how much of the drive will you pick up a signal from the transmitter?



Answer 1

Place the transmitter at the origin of the coordinate system. The equation of a circle with its center at the origin is X^2 + Y^2 = R^2 where R is the radius of the circle, which in this case is 54 miles. The coordinates of the cities are (0, 73) and (60,0).

From the city coordinates, we can write the equation of the line connecting the cities.

We can use the equation Y = mX + b, where m is the slope and b is the Y intercept.

The slope is ( 73 - 0) / (0 - 60) = -73/60. The Y intercept is 73, so the equation is Y = -73/60 X + 73.

The points of intersection of the circle and the line connecting the cities can be determined by substituting the right hand side of the straight- line equation into the Y variable in the circle equation and then solving for X.

X^2 + (-73/60 X + 73)^2 = 54^2. I will leave this for you to do as it gets pretty messy. Since the equation is quadratic in X, there are two solutions which will be the X coordinates of the points of intersection of the line with the circle. Substitute each value of X into the straight -line equation and solve for Y for each X.

The Y results will give you the Y coordinates of the points of intersection.

The length of the straight -line segment between each point of intersection is the distance for which the radio signal will be detected. The length of the straight line can be obtained using the Pythagorean theorem on the coordinates of the points of intersection. (Y2 - Y1)^2 + (X2 - X1)^2 = D^2. Solve for D, which is the distance for which the radio signal will be received.

Please check all the equations.

Answer 2

x^2 + y^2 = 54^2 is a circle with radius 54

y= -73/60)x + 73 is the line of travel

find where the line intersects the circle

then find the distance between those 2 intersection points

the points are (53.4, 8.03) and (18.22, 50.83)

distance between them = about 62.23 miles within the radio transmitter

no guarantees these calculations are error free

but it's the basic method

it helps to graph the circle and line

such as use a graphing calculator

or an online simultaneous equation solver