Question 1206903: A small radio transmitter broadcasts in a 21 mile radius. If you drive along a straight line from a city 25 miles north of the transmitter to a second city 29 miles east of the transmitter, during how much of the drive will you pick up a signal from the transmitter?
Found 2 solutions by math_tutor2020, ikleyn: Answer by math_tutor2020(3816) (Show Source):
You can put this solution on YOUR website!
Circle equation
(x-h)^2 + (y-k)^2 = r^2
where,
(h,k) = center
r = radius
Let's place the radio transmitter at the origin. Meaning h = 0 and k = 0.
The radius is r = 21 in this case.
The circle equation will then update to x^2+y^2 = 441
Points inside the circle, or on the boundary, will get the radio signal.
A = (0,25) is where you start from since it is 25 miles north of the transmitter.
B = (29,0) is where you are driving to, which is 29 miles east of the transmitter.
I'll skip a few steps, but you should find the equation of line AB is y = (-25/29)x + 25
We have this system of equations

Use substitution to plug the second equation into the first one to end up with
Skipping a few more steps, the solutions to that equation are approximately: x = 5.48588 and x = 19.24127
These are the x coordinates of the intersection points of the line and circle.
Since we're looking at approximate solutions, it seems reasonable to assume your teacher will allow you to use a graphing calculator to make quick work of this equation.
Note: The exact solutions involve very large messy numbers.
If , then points on line AB are inside the circle or on the circle's boundary.
Otherwise, you'll be outside of the circle and won't pick up the signal.
Use those x values to determine the corresponding paired y values.
x = 5.48588 leads to y = 20.2708
Let point C be located at (5.48588, 20.2708)
x = 19.24127 leads to y = 8.4127
Let point D be located at (19.24127, 8.4127)
Diagram

The diagram was made with GeoGebra which is a useful tool to verify many types of math problems.
Use the distance formula to determine these approximate segment lengths
AB = 38.28838
CD = 18.1611
Then,
CD/AB = 18.1611/38.28838 = 0.47432
For roughly 47.432% of the trip, you'll be able to pick up this particular radio signal.
Answer by ikleyn(52778) (Show Source):
|
|
|