What is the Overlapping Range of Two Radio Transmitters?

[darshanpatel]

Homework Statement

Two radio transmitters, one with a 40 mile range and one with a 60 mile range, stand 80 miles apart. you are driving 60 miles per hours on a highway parallel to the line segment connecting the two towers. How long will you be within the range of both transmitters simultaneously?

Homework Equations

-None-

The Attempt at a Solution

I know you need to find the distance where they overlap. I don't know how or what to do to find it.

I know what it looks like though.

 

[Ray Vickson]

Homework Statement

Two radio transmitters, one with a 40 mile range and one with a 60 mile range, stand 80 miles apart. you are driving 60 miles per hours on a highway parallel to the line segment connecting the two towers. How long will you be within the range of both transmitters simultaneously?

Homework Equations

-None-

The Attempt at a Solution

I know you need to find the distance where they overlap. I don't know how or what to do to find it.

I know what it looks like though.

If you know what it looks like that should be enough for you to do the question.

RGV

 

[darshanpatel]

I still don't know how to do it, it just had a picture in the book...

 

[Mark44]

How far is the road from the line that joins the two radio stations? That will affect the length of the overlap. If you could scan the picture and post it, that would help.

 

[darshanpatel]

The road is 10 miles from the segment connecting the two towers, the picture is in the book so it is kind of hard to scan, I am trying though to upload a picture...

 

[Mark44]

Sketch both circles on the same coordinate system, and write equations for both. If the center of the larger circle is at the origin, what is its equation?

Where is the center of the smaller circle? What is its equation?

Since the road is 10 miles away from the line that joins the two circles, you should be able to use the equations for the circles to find the starting and ending points of the part of the road that is in the overlap region.

 

[darshanpatel]

If the center of the larger circle is at (0,0), the equation would be x^2+y^2=3600

but would that mean that to get the equation of the smaller circle, the orgin is at (80,0) and the equation would be

x^2+y^2=1600 ?

I graphed it like this in paint: http://tinypic.com/r/a0cn5v/5

 

[Mark44]

If the center of the larger circle is at (0,0), the equation would be x^2+y^2=3600

Yes

but would that mean that to get the equation of the smaller circle, the orgin is at (80,0) and the equation would be

x^2+y^2=1600 ?

No, the origin doesn't move. What's the equation of a circle whose center is at (80, 0), and whose radius is 20?

I graphed it like this in paint: http://tinypic.com/r/a0cn5v/5

 

[darshanpatel]

with a radius of 20, the equation would be x^2+y^2=400

What do you do if the origin of a circle isn't at (0,0)? Does it make a difference?

Would the distance that overlaps be 20 miles? <-- from looking at the graph

 

[Mark44]

with a radius of 20, the equation would be x^2+y^2=400

Not if the center is at (80, 0).
Your book should have the formula for a circle of radius R whose center is at (a, b).

What do you do if the origin of a circle isn't at (0,0)? Does it make a difference?

Let's get the terms right. The question to ask is what to do if the center of the circle isn't at the origin (at (0, 0)). The origin always stays in the same place. We can move the circle around so its center is somewhere else.

Would the distance that overlaps be 20 miles? <-- from looking at the graph

You're not taking into account that the road is 10 miles away from the line that joins the two stations. Before you can figure out the length of the overlap, you need to have the equations of both circles.

 

[darshanpatel]

Ok, the complete formula for a circle is (x^2/a^2)+(y^2/b^2)=r^2
so would that make the equation x^2/1600+y^2=1600

 

[Mark44]

No, that's not it. The formula you derived, x^2/1600+y^2=1600, is not even the equation of a circle - it's the equation of an ellipse.

The equation of a circle of radius r, with its center at the point (h, k) is
(x - h)² + (y - k)² = r²

Didn't you see that one in your book?

 

[darshanpatel]

No, we have learned anything like that, I have never even seen that formula, but I do know what the (h,k) shift is from prior chapters

 

[Mark44]

No, we have learned anything like that, I have never even seen that formula, but I do know what the (h,k) shift is from prior chapters

Yes, and that's exactly what you need to use here.

 

[darshanpatel]

I don't think we will be able to use it because we have not learned it?

 

[Mark44]

You just said that you had learned about (h, k) shifts. That's exactly what's happening here. If the circle x² + y² = R² is shifted 80 units to the right, what does the equation become?

 

[darshanpatel]

the equation would be (x+80)^2+y^2=1600

or x^2+160x+6400+y^2=1600

 

[Mark44]

In my drawing, the smaller circle is centered at the origin, so its equation is x² + y² = 1600. The larger circle's equation is (x - 80)² + y² = 3600. You can set it up however you want, just as long as everything is consistent with the problem description.

In the region of overlap, find the points on the two circles where the line y = 10 intersects. This represents the road that is 10 miles away from the line that joins the two circles. Once you find the two intersection points, it should be a simple matter to figure out how long it would take to drive that distance.

 

[darshanpatel]

oh, so plug y=10 into the equations, representing distance from road, and once you find that x value, you will get two (x,y) coordinates so you can use distance formula, thank you!

 

[darshanpatel]

For the answer I got 29.82 minutes, I am wondering, because there was a lot of steps involved in solving for the ''x's" did you solve it as well? if so, did we get the same answer?

 

[Mark44]

No, I didn't solve it.