Trig question : Transmitter

Question

A small radio transmitter broadcasts in a 32 mile radius. If you drive along a straight line from a city 44 miles north of the transmitter to a second city 43 miles east of the transmitter, during how much of the drive will you pick up a signal from the transmitter?

Answer 1

It's a 44-43 sided right triangle with hypotenuse = square root of sum of 44 squared + 43 squared.

You might write the equation for a circle x squared + y squared = 32 squared

and find where it intersects the equation for the hypotenuse line with the origin as the center of the circle:

that line has y-intercept (0,44) and x intercept (43,0) slope = -44/43

equation in slope point form is: y-44 = (-44/43)(x-0)

43y-44(43) = -44x

x = -43y/44 +43

substitute that into the circle equation

and find two solutions, two points on the hypotenuse line that intersect the circle

-43y/44 + 43 all squared plus y squared = 32

That gets a little tedious in the calculations.

So, here's a here's a close estimate.

assume It's 43 miles north and 43 miles east, then the calculations simplify much more

the hypotenuse line becomes y=-x + 43

the circle equation is y² + x² = 32² =1024

substitute for y to get (-x+43)² + x² =1024

x² -86x + 1849 + x²- 1024 = 0

2x² -86x + 825 = 0

x² -43 + 412.5 = 0

x = 43/2 + (1/2)square root of (43² - 4(412.5))

x= 21.5 + or - (1/2) square root of 200

x = 21.5 + or - 7.1 = 28.6, 14.4

when x = 28.6, y=14.4, when x=14.4 y=28.6

28.6-14.4 = 12.2 = difference in x coordinates and difference in y-coordinates of the points where the circle intersects the line.

distance that you can transmit is 12,2 times square root of 2 = close to 16

the sides of the right triangle are 12.2 the hypotenuse (the part of the line within the circle of radius 32) is square root of 2 times 12.2 = about 16 or 17

It might be less, since we assumed the 44 side was 43, so it's a little more out of range

educated guess is 15

Of about 61 miles between the two cities, you can receive the transmitter for only about 15 miles

Answer 2

Equation of car:

y = -44x/43 + 44

Equation of transmitter signal

x² + y² = 32²

Determine the two points of intersection with positive x values

Determine the distance between the two points.

Compare that distance to the distance between start of travel and end of travel.