Final answer:
By using trigonometry and the cosine function, we calculate that the vertical height of the rope at a 23° angle is 3.682 meters. Subtracting the chair height, we get 0.182 meters is the part of the pole above the pivot point. The total pole height is 4.182 meters, corresponding to option (a) 4.2 meters.
Step-by-step explanation:
To determine how tall the pole of the swing ride should be, we need to solve for the vertical component of the rope's length when the swing is at a 23° tilt. First, we will consider the rope to act as a radius in a circle where the pole is the center and the chair at full swing is a point on the circumference. Using basic trigonometry, the length of the rope (hypotenuse) is 4 meters and the angle with the vertical is 23°, so the height of the pole can be calculated using the cosine function.
The vertical height (adjacent side) is given by cos(23°) × 4 meters. Subtracting the desired chair height of 3.5 meters above the ground from the total rope length will give the height of the pole above the ground.
Calculating the cosine: cos(23°) ≈ 0.9205. Multiplying by the rope length: 0.9205 × 4 meters = 3.682 meters.
To find the height of the pole, subtract the chair height (3.5 meters) from the vertical component: 3.682 meters - 3.5 meters = 0.182 meters. This is the part of the pole above the pivot point.
The total pole height is therefore the sum of the rope's vertical component and the part of the pole above the pivot point: 4 meters + 0.182 meters = 4.182 meters. Thus, the appropriate pole height is option (a) 4.2 meters.