Questions: From the lookout point on Fabrick Rock, Ann can see not only see the famous "Crooked Spire" in Chesterfield, which is 8 miles away, but also the red phone box in the village of Alton. Chesterfield and Alton are 7 miles apart. Fabrick Rock has a plaque that shows directions to famous sites, and from the plaque Ann determines that the angle between the lines to the spire and the phone box measures 19°. How far is Fabrick Rock from the phone box? (There are two possible answers.) The distance from Fabrick Rock to the phone box is miles or miles.

From the lookout point on Fabrick Rock, Ann can see not only see the famous "Crooked Spire" in Chesterfield, which is 8 miles away, but also the red phone box in the village of Alton. Chesterfield and Alton are 7 miles apart. Fabrick Rock has a plaque that shows directions to famous sites, and from the plaque Ann determines that the angle between the lines to the spire and the phone box measures 19°. How far is Fabrick Rock from the phone box? (There are two possible answers.)

The distance from Fabrick Rock to the phone box is miles or miles.

Transcript text: From the lookout point on Fabrick Rock, Ann can see not only see the famous "Crooked Spire" in Chesterfield, which is 8 miles away, but also the red phone box in the village of Alton. Chesterfield and Alton are 7 miles apart. Fabrick Rock has a plaque that shows directions to famous sites, and from the plaque Ann determines that the angle between the lines to the spire and the phone box measures $19^{\circ}$. How far is Fabrick Rock from the phone box? (There are two possible answers.) The distance from Fabrick Rock to the phone box is $\square$ miles or $\square$ miles.

Solution

Solution Steps

To solve this problem, we can use the Law of Cosines. We have a triangle with sides and an angle given: the distance from Fabrick Rock to Chesterfield (8 miles), the distance between Chesterfield and Alton (7 miles), and the angle between these two lines (19 degrees). We need to find the distance from Fabrick Rock to Alton. The Law of Cosines will help us find the unknown side of the triangle.

Step 1: Identify the Known Values

We are given a triangle with the following known values:

The distance from Fabrick Rock to Chesterfield, $ a = 8 $ miles.
The distance between Chesterfield and Alton, $ b = 7 $ miles.
The angle between these two lines, $ \angle C = 19^\circ $.

Step 2: Convert the Angle to Radians

To use the Law of Cosines, we need to convert the angle from degrees to radians: \[ \angle C_{\text{rad}} = \frac{19 \times \pi}{180} \approx 0.3316 \]

Step 3: Apply the Law of Cosines

The Law of Cosines states: \[ c^2 = a^2 + b^2 - 2ab \cos(\angle C) \] Substituting the known values: \[ c_1^2 = 8^2 + 7^2 - 2 \times 8 \times 7 \times \cos(0.3316) \] \[ c_1 = \sqrt{64 + 49 - 112 \times \cos(0.3316)} \approx 2.6649 \]

Step 4: Consider the Supplementary Angle

Since the angle could also be the supplementary angle, we calculate: \[ c_2^2 = a^2 + b^2 + 2ab \cos(\angle C) \] \[ c_2 = \sqrt{64 + 49 + 112 \times \cos(0.3316)} \approx 14.7952 \]