Final answer:
To find where the beams should be placed for the chains to be at 25 feet, we subtract the desired height from the loop's total height to get the vertical distance, then use the Pythagorean theorem to solve for the horizontal distance from the origin. The correct equation to represent this relationship is x² = 20² - 12.38², where x is the horizontal distance from the origin.
Step-by-step explanation:
The student is trying to determine the correct placement for beams that are 25 feet above the ground. To find the horizontal distance from the origin to where each beam should be placed on the rollercoaster track, we are looking for the relationship between the height of the chains and the horizontal distance from the origin at that height.
We can assume that the track forms a circle since the question implies a 360-degree loop, and the height at the top of the loop is known to be 20 m (about 65.62 feet). Since the chains need to be at a height of 25 feet, this would be at some point on the sides of the loop. If the top of the loop is the highest point, we can represent the radius of the loop as 'r' and the distance from the top of the loop down to where the beams should be placed as 'x'. This would form a right-angled triangle, with the radius as the hypotenuse and x as one of the sides. We can use the Pythagorean theorem to relate x and r.
If we subtract the 25-foot height from the total height of the loop, we are left with the vertical distance from the beam to the top of the loop, which is 65.62 feet - 25 feet = 40.62 feet (or about 12.38 meters since 1 foot = 0.3048 meters). To maintain consistency, let's convert everything into meters. Thus, r=20 m, and the vertical distance from the beam to the top is approximately 12.38 m. Now, the equation is based on the Pythagorean theorem:
Vertical distance^2 + Horizontal distance^2 = Radius^2
Hence, x^2 = r^2 - (Vertical distance)^2, which is equivalent to:
Horizontal distance (x)^2 = (20 m)^2 - (12.38 m)^2
Simplifying:
x^2 = 400 m^2 - 153.18 m^2
x^2 = 246.82 m^2
Now, expressing x in terms of a positive square root,
x = √(246.82 m^2)
Thus, each beam should be placed at a horizontal distance from the origin equal to the square root of 246.82 m^2 to ensure that the chains are fastened at the correct height of 25 feet.