Question

a surveyor estimates the length of a lake by walking 1.5 miles from point a to point b, turning at an angle of 100°, and walking 2.5 miles to point c. what is the approximate length of the lake from a to c? for help, see this worked example. 2.43 miles 2.68 miles 3.14 miles 7.20 miles

Explanation:

Step1: Identify triangle sides and angle

In $\triangle ABC$, $AB=1.5$ miles, $BC=2.5$ miles, $\angle ABC=80^\circ$.

Step2: Apply the Law of Cosines

$AC^2 = AB^2 + BC^2 - 2 \cdot AB \cdot BC \cdot \cos(\angle ABC)$
$AC^2 = 1.5^2 + 2.5^2 - 2 \cdot 1.5 \cdot 2.5 \cdot \cos(80^\circ)$
$\cos(80^\circ) \approx 0.1736$
$AC^2 = 2.25 + 6.25 - 7.5 \cdot 0.1736$
$AC^2 = 8.5 - 1.302 = 7.198$
$AC \approx \sqrt{7.198} \approx 2.68$

Answer:

2.68 miles