Final answer:
The slingshot attraction reaches a greater height at 250 feet compared to the Ferris wheel's maximum height of approximately 149 feet, providing a more thrilling experience for riders.
Step-by-step explanation:
To determine which amusement park attraction reaches a greater height, we need to analyze the given equations for the Ferris wheel and the slingshot attraction. The Ferris wheel is represented by the equation (x − 70)2 + (y − 77)2 = 5,184, which is the equation of a circle centered at (70, 77) with a radius of √5184. To find the maximum height, we look at the center and the radius. The maximum height of the Ferris wheel is therefore 77 + √5184 feet.
The slingshot attraction is modeled by the quadratic equation y = −4x2 + 40x + 50. In this form, y = ax2 + bx + c, the maximum height can be found by using the vertex formula x = −b/(2a) to find the x-coordinate of the vertex, and then substituting this back into the equation to find the maximum height (y-coordinate of the vertex). The maximum height for the slingshot attraction is thus obtained by first finding the x-coordinate: x = −(40)/(2(−4)) = 5. The maximum height is then y = −4(5)2 + 40(5) + 50 = 250 feet.
Comparing the maximum heights, we can now determine which attraction reaches a greater height. The Ferris wheel's max height is 77 + √5184, which is approximately 77 + 72 = 149 feet. Therefore, the slingshot attraction reaches a greater height at 250 feet.
In terms of context, this represents the peak experience that riders will have on each attraction, providing insight into the thrill and visual perspectives offered by the Ferris wheel versus the slingshot.