This is a projectile motion problem where a stone is thrown from the top of a building. Here is the breakdown of the given information:

• Height of the building, $H = 164 \, \text{m}$
• Initial velocity, $v_0 = 43.3 \, \text{m/s}$
• Launch angle, $\theta = 30^\circ$

You are tasked with finding the horizontal distance $x$, which is how far the stone travels horizontally before hitting the ground.

Step-by-Step Solution:

1. Resolve the velocity components: The initial velocity can be split into horizontal and vertical components. $v_{0x} = v_0 \cos(\theta) = 43.3 \cos(30^\circ)$ $v_{0y} = v_0 \sin(\theta) = 43.3 \sin(30^\circ)$

2. Time of flight: Use the vertical motion to find the total time $t$ it takes for the stone to hit the ground. The vertical displacement $y$ is given by: $y = v_{0y} t + \frac{1}{2} a t^2$ Here, $y = -164 \, \text{m}$ (since it falls 164 m), $a = -9.8 \, \text{m/s}^2$ (acceleration due to gravity), and $v_{0y}$ is the vertical component of the initial velocity.

3. Horizontal distance (Range): Once we know the time of flight $t$, we can find the horizontal distance $x$ using: $x = v_{0x} t$

Let me calculate the values for you.It seems that there was an issue with the calculation of the discriminant, which could be negative, causing a math error. This typically happens if there was an incorrect interpretation in the quadratic formula or initial setup.

Let me recheck the setup and correct the formula for vertical displacement. I will recompute.It seems like I can't do more advanced data analysis right now. Please try again later.

If you'd like, I can explain the steps in more detail or guide you through solving this manually. Let me know if you need help with any specific part of the problem.

Here are some related questions to explore:

1. How do you calculate time of flight in projectile motion problems?
2. What role does the angle of projection play in projectile motion?
3. How can you use the quadratic formula to solve time-related equations in physics?
4. How does the height of a building affect the total range in projectile motion?
5. What are the key components of velocity in two-dimensional motion?

Tip: Always check the discriminant in quadratic equations to ensure it's non-negative, otherwise, you may encounter errors.