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The problem in the image involves calculating the pressure to which a given volume of air must be compressed to fit into a spherical air tank.

### Given:
- The diameter of the spherical tank: \( 72.0 \, \text{cm} \) (which gives a radius of \( 36.0 \, \text{cm} \) or \( 0.36 \, \text{m} \)).
- The biologist estimates that \( 8700 \, \text{L} \) of air is needed for the dive.
- The final answer is required in atmospheres.

### Step 1: Volume of the spherical tank
The formula for the volume of a sphere is:
\[
V = \frac{4}{3} \pi r^3
\]
where \( r \) is the radius of the sphere.

For a radius of \( 0.36 \, \text{m} \), the volume in cubic meters is:
\[
V = \frac{4}{3} \pi (0.36)^3
\]
Let me calculate that.The volume of the spherical air tank is approximately \( 0.195 \, \text{m}^3 \).

### Step 2: Convert the tank's volume to liters
Since 1 cubic meter equals 1000 liters:
\[
V_{\text{tank}} = 0.195 \, \text{m}^3 \times 1000 = 195.4 \, \text{L}
\]

### Step 3: Use Boyle's Law
Boyle's Law relates pressure and volume for a gas at constant temperature:
\[
P_1 V_1 = P_2 V_2
\]
Where:
- \( P_1 \) is the initial pressure (1 atmosphere, since the air starts at standard pressure),
- \( V_1 \) is the initial volume of air (8700 L),
- \( P_2 \) is the final pressure (the unknown we are solving for),
- \( V_2 \) is the final volume (195.4 L, the volume of the tank).

Rearranging the formula to solve for \( P_2 \):
\[
P_2 = \frac{P_1 V_1}{V_2}
\]

Substituting values:
\[
P_2 = \frac{1 \times 8700}{195.4}
\]

Let me calculate that.The air must be compressed to approximately \( 44.5 \, \text{atm} \) to fit 8700 liters of air into the 195.4-liter spherical air tank.

Would you like any more details or have any questions?

Here are 5 related questions for practice:
1. How would the final pressure change if the volume of air required was only 5000 liters?
2. What would the final pressure be if the tank had a diameter of 1 meter?
3. How would the pressure calculation change if the temperature varied during compression?
4. How much air would fit into the tank at 10 atmospheres of pressure?
5. What is the volume of a cylindrical tank with the same radius but a height of 2 meters?

**Tip:** Always check that units are consistent, especially when converting between cubic meters and liters.

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 72.0 cm wide. The biologist estimates she will need 8700 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. Be sure your answer has the correct number of significant digits.

Learn how to calculate the pressure needed to compress 8700 L of air into a spherical air tank with a diameter of 72.0 cm. This problem uses Boyle's Law and the volume formula for spheres. Suitable for high school physics students.

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