Samantha has the opportunity to win $1 million by sinking a half court shot. She shoots the basketball with a speed of 69.2 ft/s while standing 50 ft away from the basketball hoop. The hoop is 10 ft above the floor and the basketball leaves her hand 6.7 ft above the floor. What 2 angles can she shoot at?

To solve this, we'll model the problem using the equations of projectile motion. The horizontal and vertical components of the basketball's motion must satisfy the following conditions:

- The horizontal distance traveled is 50 ft (the distance from Samantha to the hoop).
- The vertical distance traveled is the difference between the hoop's height (10 ft) and the release height (6.7 ft), which is $ \Delta y = 10 - 6.7 = 3.3 $ ft.
- The initial speed is 69.2 ft/s.

### Step 1: Define the equations of motion
We'll split the motion into horizontal and vertical components:

1. **Horizontal motion** (constant velocity):
   \[
   x(t) = v_x t = v_0 \cos(\theta) t
   \]
   where:
   - $ v_x $ is the horizontal velocity component,
   - $ v_0 = 69.2 \, \text{ft/s} $ is the initial speed,
   - $ \theta $ is the launch angle,
   - $ t $ is the time of flight,
   - $ x = 50 \, \text{ft} $ is the horizontal distance to the hoop.

2. **Vertical motion** (constant acceleration due to gravity):
   \[
   y(t) = y_0 + v_y t - \frac{1}{2} g t^2
   \]
   where:
   - $ y_0 = 6.7 \, \text{ft} $ is the initial height of the ball,
   - $ v_y = v_0 \sin(\theta) $ is the vertical velocity component,
   - $ g = 32.2 \, \text{ft/s}^2 $ is the acceleration due to gravity,
   - $ y(t) = 10 \, \text{ft} $ is the final height at the hoop.

### Step 2: Solve for time $ t $
From the horizontal equation:
\[
t = \frac{x}{v_0 \cos(\theta)} = \frac{50}{69.2 \cos(\theta)}
\]

### Step 3: Plug into the vertical equation
Substitute this expression for $ t $ into the vertical motion equation:
\[
10 = 6.7 + (69.2 \sin(\theta)) \left( \frac{50}{69.2 \cos(\theta)} \right) - \frac{1}{2} (32.2) \left( \frac{50}{69.2 \cos(\theta)} \right)^2
\]

Simplifying this will give us an equation in terms of $ \theta $ that we can solve for the two angles.

### Step 4: Solve the quadratic equation
After simplifying and solving the resulting quadratic equation, we can find the two possible values for $ \theta $.

I'll perform these calculations now.It seems there was an issue with mixing symbolic math with numerical values. I'll correct that and use a fully symbolic approach to solve for the angles $\theta$. Let me retry the calculation using a better approach for handling the equation symbolically.It seems like I can’t do more advanced data analysis right now. Please try again later.

In the meantime, feel free to ask any other questions, or if you'd like, I can guide you through how to solve the problem manually. Let me know!

Here are five questions that can extend from this problem:
1. How does air resistance affect the trajectory of the basketball?
2. What happens if Samantha shoots from a different distance?
3. How do different release heights affect the possible angles?
4. How would changing the initial speed of the ball impact the angles?
5. Can you solve for the time of flight for both angles?

**Tip:** Understanding the relationship between angles and initial velocity is key to mastering projectile motion problems.

Learn how to solve a projectile motion problem involving a half-court basketball shot. Samantha shoots from 50 ft away with a speed of 69.2 ft/s. This problem explores key concepts such as horizontal and vertical motion, using kinematic equations to find the two possible launch angles.

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