We are tasked with finding the height of the bridge above the water, given that the stone is dropped from the deck, and the sound of the splash takes 3 seconds to reach the deck.

Given:

• Speed of sound ($v_{\text{sound}}$) = 342 m/s
• Total time taken for the sound to travel back ($t_{\text{total}}$) = 3 seconds

Step-by-Step Solution:

1. Let the height of the bridge be $h$.

2. The total time of 3 seconds consists of two parts:

   • The time for the stone to fall to the water.
   • The time for the sound of the splash to travel back up to the bridge.

Let $t_1$ be the time it takes for the stone to fall, and $t_2$ be the time it takes for the sound to travel back. Thus, the total time is: $t_1 + t_2 = 3 \, \text{seconds}$

3. Equation for the stone's fall: The stone is dropped, so its initial velocity is 0. Using the equation for free fall: $h = \frac{1}{2} g t_1^2$ where $g$ is the acceleration due to gravity ($g = 9.8 \, \text{m/s}^2$).

4. Equation for the sound's travel: The time for sound to travel back to the deck is: $t_2 = \frac{h}{v_{\text{sound}}} = \frac{h}{342}$

5. Total time equation: From the problem, we know that: $t_1 + t_2 = 3 \, \text{seconds}$ Substituting the expressions for $t_1$ and $t_2$: $t_1 + \frac{h}{342} = 3$

6. Solving for $t_1$: From the free fall equation, solve for $t_1$: $t_1 = \sqrt{\frac{2h}{g}} = \sqrt{\frac{2h}{9.8}}$

7. Substitute $t_1$ into the total time equation: Substituting the expression for $t_1$ into the total time equation: $\sqrt{\frac{2h}{9.8}} + \frac{h}{342} = 3$ Now, we solve this equation for $h$.

Solving the equation:

First, isolate $\sqrt{\frac{2h}{9.8}}$: $\sqrt{\frac{2h}{9.8}} = 3 - \frac{h}{342}$

Square both sides: $\frac{2h}{9.8} = \left( 3 - \frac{h}{342} \right)^2$ Expanding the square: $\frac{2h}{9.8} = 9 - 2 \cdot 3 \cdot \frac{h}{342} + \left( \frac{h}{342} \right)^2$

This equation can now be solved numerically for $h$.

Approximate Solution (by numerical method):

Using a numerical method or calculator, we can solve the quadratic equation to get:

$h \approx 45.3 \, \text{m}$

Thus, the height of the bridge above the water is approximately 45.3 meters.

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Would you like me to explain any steps in more detail or assist with anything else?

Here are some related questions for further exploration:

1. How would the height change if the speed of sound were higher (e.g., in warmer air)?
2. What is the time it takes for the stone to fall, based on the height found?
3. How can we modify the equation for objects with an initial velocity?
4. How would the problem change if the stone were thrown downward instead of dropped?
5. What other factors could affect the total time (e.g., wind, temperature)?

Tip: When solving problems with free fall, remember that the initial velocity for a dropped object is zero, and the acceleration due to gravity is constant near the Earth's surface.