At sea level the water pressure is equal to the air pressure, which is approximately 1

atmosphere (atm). As you dive down, the pressure increases about 1 atm for every 10m you descend.

a. Find an equation that relates pressure, p, in atmospheres, to depth below water surface, d, in meters.

b. What do the slope and p-intercept mean in context?

c. Most scuba gear and regular certifications allow you to dive to about 40m below the surface of the water. What

pressure is your body feeling at that depth?

Let's break down the problem step by step:

### Part (a): Finding the equation relating pressure \( p \) to depth \( d \)

Given:
- At the water surface (\( d = 0 \) meters), the pressure \( p = 1 \) atmosphere (atm).
- The pressure increases by 1 atm for every 10 meters of depth.

We can use the following information to establish a linear relationship between pressure and depth:
- Slope: The pressure increases by 1 atm for every 10 meters, so the slope \( m = \frac{1 \text{ atm}}{10 \text{ m}} = 0.1 \) atm/m.
- The equation of a line in slope-intercept form is \( p = md + b \), where \( p \) is the pressure, \( d \) is the depth, \( m \) is the slope, and \( b \) is the y-intercept.

Since at \( d = 0 \), \( p = 1 \) atm, the y-intercept \( b = 1 \).

**Equation:**
\[
p = 0.1d + 1
\]

### Part (b): Interpreting the slope and p-intercept

- **Slope (0.1 atm/m):** This means that for every 10 meters you descend, the pressure increases by 1 atmosphere. It's the rate at which pressure increases with depth.

- **p-intercept (1 atm):** This is the pressure at the surface of the water (at \( d = 0 \) meters). It represents the atmospheric pressure at sea level.

### Part (c): Calculating the pressure at 40 meters below the surface

Using the equation \( p = 0.1d + 1 \):

For \( d = 40 \) meters:
\[
p = 0.1(40) + 1 = 4 + 1 = 5 \text{ atm}
\]

**Answer:**
At a depth of 40 meters, the pressure your body feels is **5 atmospheres (atm)**.

Would you like further details or have any questions?

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Here are 8 related questions that expand on this information:

1. How would the pressure change at a depth of 60 meters?
2. What would be the pressure at 100 meters below the surface?
3. If the pressure at 20 meters is 3 atm, how deep would you be for a pressure of 6 atm?
4. What happens to the pressure if you ascend 5 meters from a depth of 25 meters?
5. How does the pressure equation change if the water density varies?
6. What is the physical significance of water pressure on a diver's body at different depths?
7. How can divers adjust their gear to handle different pressures at various depths?
8. What is the relationship between pressure and volume in a diver's air tank as they descend?

**Tip:** Understanding how pressure changes with depth is crucial for divers to avoid issues like decompression sickness and barotrauma.

Explore how water pressure changes with depth using linear equations. Learn about the slope and intercept meanings in context. Calculate pressure at specific depths and understand implications for scuba diving. Ideal for students and enthusiasts of physics and diving.

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