Answer: He will pick up a signal from the transmitter 29.43 miles of the drive.
Step-by-step explanation:
Since we have given that
Radius = 29 mile
So, equation of circle would be
[tex]x^2+y^2=29^2=841[/tex]
There is a straight line from a city 32 miles north i.e. (0,32) to a second city 40 miles east of the transmitter i.e. (40,0)
So, equation of line would be
[tex]y-32=\dfrac{0-32}{40-0}(x-0)\\\\y-32=\dfrac{-32}{40}x\\\\y-32=-0.8x\\\\y+0.8x=32\\\\y=32-0.8x[/tex]
So, we will put this value in the equation of circle.
[tex]x^2+(32-0.8x)^2=841\\\\x^2+1024+0.64x^2-51.2x=841\\\\1.64x^2-51.2x=841-1024\\\\1.64x^2-51.2x=-183\\\\1.64x^2-51.2x+183=0[/tex]
So, x= 4.12 and x = 27.1
So, y = 28.7, y = 10.32
So, distance between them would be
[tex]\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\\\\=\sqrt{(27.1-4.12)^2+(28.7-10.32)^2}\\\\=29.43[/tex]
Hence, he will pick up a signal from the transmitter 29.43 miles of the drive.
