Question

question 7 of 10
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. 126 feet
b. 114 feet
c. 41 feet
d. 25 feet

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines formula is \(c^{2}=a^{2}+b^{2}-2ab\cos C\). Let \(a = 60\), \(b = 80\), and \(C=30^{\circ}\). Then \(c^{2}=60^{2}+80^{2}-2\times60\times80\times\cos30^{\circ}\).

Step2: Calculate each term

• \(60^{2}=3600\), \(80^{2}=6400\), and \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\approx0.866\).
• \(2\times60\times80\times0.866 = 2\times4800\times0.866=9600\times0.866 = 8313.6\).
• \(a^{2}+b^{2}=3600 + 6400=10000\).
• \(c^{2}=10000-8313.6 = 1686.4\).

Step3: Find \(c\)

Take the square - root of \(c^{2}\), \(c=\sqrt{1686.4}\approx41\).

Answer:

C. 41 feet