Question

the town of asherville is 18 miles away from the town of bakersfield, which is 24 miles away from the town of chaplain.

what is the approximate distance between chaplain and asherville?

29.8 miles
30.9 miles
14.7 miles
16.4 miles

Explanation:

Identify the given values and unknown

Using the Law of Cosines knowledge point
Let \(A\) represent Asherville, \(B\) represent Bakersfield, and \(C\) represent Chaplain.
Given:

• Side \(c = AB = 18\text{ miles}\)
• Side \(a = BC = 24\text{ miles}\)
• Angle \(A = 94^\circ\)
• Unknown side \(b = AC\) (distance between Chaplain and Asherville)

Set up the Law of Cosines equation

Using the Law of Cosines knowledge point

$$ a^2 = b^2 + c^2 - 2bc \cos(A) $$

Substitute the known values:

$$ 24^2 = b^2 + 18^2 - 2 \cdot b \cdot 18 \cdot \cos(94^\circ) $$

Simplify the quadratic equation

Using the Law of Cosines knowledge point

$$
LATEXBLOCK0
$$

Solve for the positive distance

Using the Law of Cosines knowledge point

$$
LATEXBLOCK1
$$

Answer:

• 29.8 miles
• 30.9 miles
• 14.7 miles (Correct answer)
• 16.4 miles