A swing hangs from a beam 10 feet high, with the seat hanging 3 feet above the ground. In motion, the swing moves back and forth from −135° to −45° to the horizontal. How far forward and back does the swing move along the ground at the extremes? If necessary, round your answers to the nearest hundredths.

9.9 feet.

Step-by-step explanation:

Solution:

- If the swing were to hang straight down, it would be hanging at an angle of -90 degrees from the horizontal. If it is moving from -45 degrees from the horizontal to -135 degrees from the horizontal, this means that it is swinging from 45 degrees to the right of the straight-down position to 45 degrees to the left of the straight-down position.

- The swing is of length L = 10 - 3 = 7 feet. Recall that it hangs from a beam 10 feet above the ground and the seat hangs 3 feet above the ground in its straight-down rest position.

- Let's consider the swing at one of its extreme positions where it makes an angle of 45 degrees from the vertical. In this position, you can imagine the swing forming a right triangle with the highest vertex angle being 45 degrees and the hypotenuse being L = 7 feet. To calculate the horizontal leg of this triangle, you would use the sine function.

                                     sin(45 deg) = x/L

So,

                                    x = L*sin(45 deg)

                                    x = 7*sin(40 deg) = 4.95 feet

- But, this is only the horizontal distance that the swing traverses from one extreme position to hanging straight down. It needs to complete its motion and swing up to the other extreme position. So, in moving from one extreme position to the other, the swing traverses a horizontal distance of    

                                     2x = 2*4.95 feet = 9.9 feet.