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A small radio transmitter broadcasts in a 27 mile radius. If you drive along a straight line from a city 35 miles north of the transmitter to a second city 31 miles east of the transmitter, during how much of the drive will you pick up a signal from the transmitter?

 Oct 22, 2019
 #1
avatar+104969 
+2

Let the transmitter be located at (0,0)

 

The equation for its effective coverage area will be a circle  centered at  (0,0)  with a radius  of  27

So....the equation is   

x^2 + y^2  = 27^2

x^2 + y^2  = 729

 

Let the point 35  miles north of the transmitter be  (0, 35)

Let the point 31  miles east of the trasmitter   be (31,0)

 

 

 

The line connecting these (the line of travel)  will have a slope  of  [ 35 - 0] / [ 0 - 31]  =  -35/31

 

The equation of this line will be

 

y  = (-35/31)x  + 35

 

We could solve this  algebraically, but a graph seems easier to deal with  

 

See here : https://www.desmos.com/calculator/ksd2p5jdwl

 

The signal will be received  betwen the points  (8.221 , 25.718)  and (26.523, 5.055)

 

The distance between these points  is  

 

√ [ ( 26.523 - 8.221)^2  + ( 25.718 - 5.055)^2 ]  ≈  27.6 miles     (1)

 

The distance  between  (0, 35)  and (31,0)  =  √ [ ( 31)^2  + ( 35)^2 ]  ≈  46.8  miles  (2)

 

So....you will receive the signal  about  (1)/ (2)    =  27.6 / 46.8  ≈  59%  of the drive

 

 

cool cool cool

 Oct 22, 2019
edited by CPhill  Oct 22, 2019
 #2
avatar+6 
+1

Sorry for the bother but is there any chance you could show me how to do it algebraically?

It's review for a test and I won't have acess to a graphing calculator.

evanborot  Oct 22, 2019
 #3
avatar+104969 
+1

Yeah....I can work it out for you.....in just a few....

 

 

cool cool cool

CPhill  Oct 22, 2019
 #4
avatar+104969 
+2

We need to find the  x intersection points  of

 

x^2 + y^2   = 729     and

 

y = (-35/31)x + 35

 

So....subbing the second equation into the first for y  we have that

 

x^2  +  [ (-35/31)x + 35]^2  = 729     

 

x^2  +  1225x^2/961  -2450x/31  + 1225  = 729

 

[961 + 1225]x^2 / 961 - 2450x/31  + 496  = 0     multiply through by 961

 

2186  x^2  -  75950x  + 476656  =  0

 

Using the quadratic formula    we have that 

 

75950  ±√ [ (75950^2 - 4(2186)(476656) ]

_____________________________________

         2 *  2186

 

 

Evaluating this we get that  x =  8.221  and  x = 26.523

 

Putiing the first value into  the linear equation we have that

 

y = (-35/31)(8.221) + 35   ≈ 25.718

 

Putting the second vallue into the linear eqation we have that

 

y = (-35/31)(26.523) + 35  ≈ 5.055

 

So....the intersection points of the circle and line of travel  are ( 8.221, 25.718)  and (25.523, 5.055)

 

And  I just used the distance formula twice to solve the rest of the problem

 

cool cool cool

 Oct 22, 2019

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