Question

The Stratosphere Tower in Las Vegas is 921 feet tall and has a "needle" at its top that extends even higher into the air. A thrill ride called the Big Shot catapults riders 160 feet up the needle and then lets them fall back to the launching pad.

a. The height h (in feet) of a rider on the Big Shot can be modeled by
h = 16t² + v₀ t + 921 where t is the elapsed time (in seconds) after launch and v0 is the initial vertical velocity (in feet per second). Find v0 using the fact that the maximum value of h is 921 + 160 = 1081 feet.
b. A brochure for the Big Shot states that the ride up the needle takes 2 seconds. Compare this time with the time given by the model h = 16t² + v₀ t + 921 where v0 is the value you found in part (a). Discuss the model’s accuracy."

Answer 1

The initial vertical velocity (v₀) of the ride can be found by setting the maximum height equation equal to the height reached by the ride and solving for v₀. The value of v₀ is 10 feet per second. The model's accuracy in terms of predicting the time it takes for the rider to reach the maximum height is confirmed by the fact that the brochure's stated time matches the time found in the model.

Step-by-step explanation:

To find the initial vertical velocity (v0), we can use the fact that the maximum height (h) reached by the rider is 921 + 160 = 1081 feet. The equation that models the height of the rider is h = 16t² + v₀ t + 921, where t is the time elapsed after launch. At the maximum height, the velocity is 0, so we substitute h = 1081 and solve for v0.

1081 = 16t² + v₀ t + 921

Subtracting 921 from both sides gives: 160 = 16t² + v₀ t

This equation can be rearranged into the quadratic equation form: 16t² + v₀ t - 160 = 0

Using the quadratic formula, we can solve for t. The positive solution will represent the time it takes for the rider to reach the maximum height: t = (-v₀ ± √(v₀² + 4(16)(160)))/ (2(16))

Since t = 2 seconds, we can substitute this value into the equation and solve for v₀: 2 = (-v₀ ± √(v₀² + 4(16)(160)))/(32)

Simplifying the equation gives us two possible values for v₀: -5 and 10. Since the initial vertical velocity should be positive, we can conclude that v₀ = 10 feet per second.

b. The brochure states that the ride up the needle takes 2 seconds. This matches the time we found in part a, indicating that the model is accurate in terms of predicting the time it takes for the rider to reach the maximum height.

However, it's important to note that the model assumes no other external factors such as air resistance or friction. In reality, these factors may affect the accuracy of the model. Nevertheless, for the given conditions, the model provides a reasonably accurate prediction of the height and time for the Big Shot ride.