A park ranger at point A observes a fire in the direction $\mathrm { N } 27 ^ { \circ } 27 ^ { \prime } \mathrm { E }$ . Another ranger at point B, 9 miles due east of A, sites the same fire at $\mathrm { N } 54 ^ { \circ } 28 ^ { \prime } \mathrm { W }$ . Determine the Distance from point B to the fire. Round answer to two decimal places.
A) 4.19 miles
B) 4.66 miles
C) 8.07 miles
D) 7.40 miles
E) 8.22 miles

Question

Quizplus · Accepted Answer

1. Draw a diagram:

[asy]
pair A, B, C;

A = (0,0);
B = (9,0);
C = (4.66,4.19);

draw(A--C--B);
draw(rightanglemark(A,C,B,15));

label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,N);
[/asy]

2. Calculate the angles:

$$\begin{aligned} \angle ACB &= 180^\circ - \angle BCA - \angle CAB \ &= 180^\circ - (54^\circ 28' - 27^\circ 27') - 27^\circ 27' \ &= 72^\circ 55' \end{aligned}$$

$$\angle ABC = 180^\circ - 90^\circ - \angle ACB = 17^\circ 05'$$

3. Use the Law of Sines to find the length of BC:

$$\frac{BC}{\sin \angle ACB} = \frac{AB}{\sin \angle ABC}$$

$$BC = \frac{AB \sin \angle ACB}{\sin \angle ABC} = \frac{9\sin 72^\circ 55'}{\sin 17^\circ 05'} \approx 8.07$$

Therefore, the answer is (C) 8.07 miles.

A Park Ranger at Point a Observes a Fire in the Direction