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Let's break down the problem step by step:

### Problem Summary:
- A **spherical air tank** has a diameter of **64.0 cm**.
- The biologist needs **5400 L** of air.
- We need to calculate the **pressure** (in atmospheres) to which this volume must be compressed to fit into the tank.

### Step 1: Calculate the volume of the spherical air tank
The volume \( V \) of a sphere is given by the formula:
\[
V = \frac{4}{3} \pi r^3
\]
where \( r \) is the radius of the sphere. The diameter is 64.0 cm, so the radius is:
\[
r = \frac{64.0 \, \text{cm}}{2} = 32.0 \, \text{cm}
\]
We should convert the radius to meters (since 1 L = 1000 cm³):
\[
r = 0.32 \, \text{m}
\]
Now, calculate the volume of the air tank in cubic meters:
\[
V = \frac{4}{3} \pi (0.32)^3
\]

### Step 2: Use the Ideal Gas Law to relate pressure and volume
The relationship between pressure and volume for a fixed amount of gas is described by Boyle's Law:
\[
P_1 V_1 = P_2 V_2
\]
Where:
- \( P_1 \) is the initial pressure (1 atm, since the air is initially at atmospheric pressure).
- \( V_1 = 5400 \, \text{L} \) (or 5.4 m³, since \( 1 \, \text{L} = 0.001 \, \text{m}^3 \)).
- \( P_2 \) is the final pressure (what we need to find).
- \( V_2 \) is the volume of the spherical tank (in m³, from step 1).

Let's calculate the volume of the tank and solve for \( P_2 \).The volume of the spherical air tank is approximately **0.137 m³**. 

To fit **5400 L** (or 5.4 m³) of air into this tank, the air must be compressed to a pressure of approximately **39.3 atmospheres**.

Would you like more details or have any questions? Here are 5 related questions to consider:
1. How does Boyle's Law apply to real-life scenarios, like scuba diving?
2. What is the relationship between volume and pressure in gases?
3. How would the pressure change if the tank had a different size?
4. What units are commonly used for measuring pressure in different fields?
5. How does temperature affect the compression of gases?

### Tip: When dealing with large pressure changes, always account for the temperature to avoid errors in real-world applications!

A marine biologist is preparing a deep-sea submersible for a dive. The sub stores breathing air under high pressure in a spherical air tank that measures 64.0 cm wide. The biologist estimates she will need 5400 L of air for the dive. Calculate the pressure to which this volume of air must be compressed in order to fit into the air tank. Write your answer in atmospheres.

Learn how to calculate the pressure needed to compress 5400 L of air into a spherical tank with a 64.0 cm diameter using Boyle's Law. Understand how volume and pressure relate for gases, with step-by-step breakdowns for high school students in Grades 10-12.

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