Question

note: diagram not drawn to scale. select the correct answer. to estimate the length of a pond, a biologist walks 1,200 feet from point x to point y, then turns 110° and walks 1,500 feet to point z. what is the approximate length of the pond?

Explanation:

Step1: Identify the law of cosines formula

The law of cosines for a triangle with sides \(a\), \(b\), \(c\) and the angle \(\theta\) opposite to side \(c\) is \(c^{2}=a^{2}+b^{2}-2ab\cos\theta\). Here, let \(a = 1200\), \(b=1500\), and \(\theta = 110^{\circ}\).

Step2: Calculate the values for the formula

First, find \(\cos(110^{\circ})\approx - 0.342\). Then, \(a^{2}=1200^{2}=1440000\), \(b^{2}=1500^{2}=2250000\), and \(2ab = 2\times1200\times1500=3600000\).

Step3: Substitute values into the law - of - cosines formula

\(c^{2}=1440000 + 2250000-3600000\times(-0.342)\).
\(c^{2}=1440000+2250000 + 1231200\).
\(c^{2}=4921200\).

Step4: Find the value of \(c\)

\(c=\sqrt{4921200}\approx2218\) (rounded to the nearest whole number).

Answer:

Approximately 2218 ft