Final answer:
Using trigonometry and the cosine function, the new height of the wrecking ball is calculated by multiplying the length of the cable (5.0 meters) by the cosine of 30 degrees and subtracting that from the total length of the cable. The answer is approximately 4.3 meters, option c.
Step-by-step explanation:
To calculate the new height of the wrecking ball when the cable makes a 30-degree angle with the vertical, we can use trigonometric principles. Specifically, we'll use the cosine function because it relates the adjacent side (height above the lowest point) to the hypotenuse (the length of the cable) in a right-angled triangle.
The formula we use is:
Adjacent side (height above lowest point) = Hypotenuse (length of cable) × Cos(angle)
The length of the cable is given as 5.0 meters. The adjacent side will be the vertical distance from the lowest point that the ball is pulled back to. Thus, we can calculate:
Height above lowest point = 5.0 meters × cos(30°)
We can find the cosine of 30 degrees using a calculator or a trigonometric table, which is approximately 0.866.
So, the calculation is:
Height above lowest point = 5.0 meters × 0.866 ≈ 4.33 meters
However, we are looking for the new height of the ball, which is the distance from the highest point the ball can reach (the length of the cable) minus the vertical distance from the lowest point it was pulled back to.
New height of the ball = Cable length - Height above lowest point
New height of the ball = 5.0 meters - 4.33 meters
New height of the ball = 0.67 meters
This new height represents how far the ball has been lifted from its lowest point. Since we are looking for the height from the original (starting) position, we take the total length of the cable and subtract the calculated distance.
New height of the ball from the starting position = 5.0 meters - 0.67 meters
New height of the ball from the starting position = 4.33 meters
Therefore, the new height of the ball from the starting position is approximately 4.3 meters, which corresponds to option c: 4.3 meters.