trigonometry circle

Question

A small radio transmitter broadcasts in a 59 mile radius. If you drive along a straight line from a city 74 miles north of the transmitter to a second city 77 miles east of the transmitter, during how much of the drive will you pick up a signal from the transmitter?

Answer 1

A small radio transmitter broadcasts in a 59 mile radius. If you drive along a straight line from a city 74 miles north of the transmitter to a second city 77 miles east of the transmitter, during how much of the drive will you pick up a signal from the transmitter?

a=74

b=77

I drew this on my cad. The rad=59"= each side. The angle is 50.78°

Since there's 2 right triangles; the formula to use is a^2+b^2=c^2 and a/sinα=b/sinβ=c/sinγ.

α=25.39° (50.78/2)

β=64.61° (α+β)=90

γ=90°

b=59

So: a/sin25.39=59 (angle of 90°=1).

a=25.2978

* 2

=50.5957.......... miles

Answer 2

Equation of strait line is: x/77 + y/74 = 1 and circle where works radio is x² + y² = 59². We have to solve this system to get points of intersection.

y = 74(1 - x/77); x² + 74²(1 - 2x/77 + x²/77²) = 59²; (1 + 74²/77²)x² - 2·74²x/77 + 74² - 59² = 0

11405x² - 843304x + 11828355 = 0

x² - 73.942x + 1037.12 = 0, x₁ = 18.812 mi, x₂ = 55.130 mi, y₁ = 55.921; y₂ = 21.018.

Now to find solution we have to evaluate distance between points (18.812, 55.921) and (55.130, 21.018)

d = 50.371 mi

Answer 3

This is a numerical nuisance...but the logic is easy enough.

The 2 cities have co-ordinates (0,74) and (77,0).

First compute the equation of the straight line through them.

The broadcast range has the equation y=sqrt(59² - x²).

The intersection points of the line and the circle can be found by solving the 2 equations as a simultaneous pair.

Then the distance between the points is the distance during which there will be radio reception.

Graphically, the points are approximately (55.13,21.02) and (18.81,55.92).