Question

sitting on a park bench, you see a swing that is 60 feet away and a slide that is 80 feet away. the angle between them is 30°. what is the approximate distance between the swing and the slide?
a. 25 feet
b. 114 feet
c. 41 feet
d. 126 feet

Explanation:

Step1: Identify the law of cosines formula

Let \(a = 60\), \(b = 80\), and \(\theta=30^{\circ}\). The law of cosines formula for finding the side \(c\) in a triangle is \(c^{2}=a^{2}+b^{2}-2ab\cos\theta\).

Step2: Substitute the values

Substitute \(a = 60\), \(b = 80\), and \(\theta = 30^{\circ}\) into the formula. \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\approx0.866\). So \(c^{2}=60^{2}+80^{2}-2\times60\times80\times\cos30^{\circ}\).
\[

$$\begin{align*} c^{2}&=3600 + 6400-9600\times0.866\\ c^{2}&=3600 + 6400 - 8313.6\\ c^{2}&=10000-8313.6\\ c^{2}&=1686.4 \end{align*}$$

\]

Step3: Find \(c\)

Take the square - root of \(c^{2}\). \(c=\sqrt{1686.4}\approx 41\).

Answer:

C. 41 feet