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The image presents an application of quadratic equations to solve a real-world problem involving a water balloon dropped from a height. Let's go through the problem and solution step-by-step.

### Problem:
Tom wants to drop a water balloon from the 6th floor of a building, which is 70 feet high, to hit his friend standing below. His friend is 6 feet tall. The question is to find out how many seconds it will take for the water balloon to hit his friend.

### Solution:
1. **Initial Setup**:
   - The building height is \( 70 \) feet, and the friend's height is \( 6 \) feet.
   - The water balloon will be in free fall for \( 70 - 6 = 64 \) feet.

2. **Equation of Motion**:
   The equation given is:
   \[
   h = -16t^2 + 70
   \]
   Here, \( h \) represents the height of the balloon at time \( t \). The value \( -16t^2 \) comes from the formula for the distance fallen under gravity in feet (using acceleration \( g \approx 32 \text{ ft/s}^2 \), and dividing by 2 gives \( 16t^2 \)).

3. **Solving for When the Balloon Hits the Friend**:
   Since the balloon falls 64 feet, we need to set \( h = 6 \) feet:
   \[
   6 = -16t^2 + 70
   \]
   Rearrange to isolate \( t \):
   \[
   -16t^2 = 6 - 70
   \]
   \[
   -16t^2 = -64
   \]
   \[
   t^2 = 4
   \]
   \[
   t = \sqrt{4} = 2 \text{ seconds}
   \]

4. **Rejecting Negative Time**:
   In the solution, only the positive value of \( t \) is relevant because time cannot be negative. Therefore:
   \[
   t = \sqrt{2} \approx 1.4141 \text{ seconds}
   \]

### Explanation:
- The equation models the motion of the water balloon under gravity.
- We calculated the time it takes for the balloon to fall a certain distance (from the window to the top of the friend’s head) using basic kinematics.

Would you like me to explain anything further or break down the calculations differently?

### Related Questions:
1. What is the general formula for the height of an object in free fall?
2. How can we derive the value \( -16t^2 \) for the motion under gravity?
3. What changes in the equation if the water balloon was thrown downwards instead of just being dropped?
4. How would the time change if the friend moved further away from the building?
5. How can we find the velocity of the balloon when it hits the friend?

**Tip:** 
When solving motion problems, always start by identifying all given values and determining which equation of motion applies.

Tom wants to prank his friend by hitting him with a water balloon on a hot summer day. If the balloon is dropped from the 6th floor of an apartment building that is 70 feet high, after how many seconds will the balloon hit the friend? Assume the friend is 6 feet tall.

Learn how to apply quadratic equations to solve real-world problems like free fall motion. This example explores how long it takes for a water balloon dropped from 70 feet to hit a friend standing below. The calculation uses the formula for free fall motion, simplifying it into a quadratic equation. Suitable for students in Grades 9-12.

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