Trigonometry Question

Question

A reservoir supplies water through an aqueduct to Springfield, which is 15 miles from the reservoir at an 25º angle southeast of the reservoir. A pumping station ad Springfield pumps water to Centerville, which is 7.5 miles due east of Springfield. Plans have been made to build an aqueduct directly from the reservoir to Centerville. How long will the new aqueduct be?

Answer 1

Answerr:

The new aqueduct from the reservoir to Centerville would be 22.03 miles.

Answer 2

Law of sines says:

sin 25/x = sin t/7.5 = sin(155-t)/15

cross multiplies the last two proportions:

15t = 7.5 sin(155-t)

15t = 7.5 sin155 cos t - 7.5cos155 sin t

Let f(t) = 7.5 sin155 cos t - 7.5cos155 sin t - 15t

155 degrees = (155/180) pi = 31/36 * pi

the approximate solution is t=0.356 radians =

20.397297506657306232140143113821 degrees

then sin 25/x = sin t/7.5

cross multiplies:

7.5 * sin 25 = x * sin t

7.5 * sin 25/ sin t = x

x = 9.0943580957748345596302944036303