Sitting on a park bench, you see a swing that is 100 feet away and a slide that is 60 feet away. The angle between them is 30°. What is the approximate distance between the swing and the slide?A.53 feetB.57 feetC.147 feetD.120 feet
Question
Sitting on a park bench, you see a swing that is 100 feet away and a slide that is 60 feet away. The angle between them is 30°. What is the approximate distance between the swing and the slide?A.53 feetB.57 feetC.147 feetD.120 feet
Solution
To solve this problem, we can use the law of cosines. The law of cosines is a formula used in geometry to find a side of a triangle when the lengths of the other two sides and the angle between them are known. The formula is:
c² = a² + b² - 2ab*cos(C)
where:
- c is the length of the side opposite angle C
- a and b are the lengths of the other two sides
- C is the angle between sides a and b
In this case:
- a is the distance to the swing, which is 100 feet
- b is the distance to the slide, which is 60 feet
- C is the angle between the swing and the slide, which is 30 degrees
Substituting these values into the formula gives:
c² = 100² + 60² - 210060*cos(30)
Calculating this gives:
c² = 10000 + 3600 - 12000*cos(30)
cos(30) is equal to √3/2, so:
c² = 10000 + 3600 - 12000*√3/2
c² = 13600 - 10392.3
c² = 3207.7
Taking the square root of both sides gives:
c = √3207.7
c ≈ 56.7 feet
So, the distance between the swing and the slide is approximately 57 feet. Therefore, the correct answer is B. 57 feet.
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