Question

A small radio transmitter broadcasts in a 29 mile radius. If you drive along a straight line from a city 32 miles north of the transmitter to a second city 40 miles east of the transmitter, during how much of the drive will you pick up a signal from the transmitter

Answer 1

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

$x^2+y^2=29^2=841$

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

$y-32=\dfrac{0-32}{40-0}(x-0)\\\\y-32=\dfrac{-32}{40}x\\\\y-32=-0.8x\\\\y+0.8x=32\\\\y=32-0.8x$

So, we will put this value in the equation of circle.

$x^2+(32-0.8x)^2=841\\\\x^2+1024+0.64x^2-51.2x=841\\\\1.64x^2-51.2x=841-1024\\\\1.64x^2-51.2x=-183\\\\1.64x^2-51.2x+183=0$

So, x= 4.12 and x = 27.1

So, y = 28.7, y = 10.32

So, distance between them would be

$\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\\\\=\sqrt{(27.1-4.12)^2+(28.7-10.32)^2}\\\\=29.43$

Hence, he will pick up a signal from the transmitter 29.43 miles of the drive.