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The problem involves calculating the pressure at which a volume of air needs to be compressed in a spherical air tank, based on Boyle's Law. Here's how we can approach it:

### Given Information:
1. **Tank diameter:** 89.0 cm.
2. **Required air volume:** 9000 L.
3. **Boyle's Law:** \( P_1 V_1 = P_2 V_2 \) (where \( P \) is pressure and \( V \) is volume).
4. **Initial pressure:** \( P_1 = 1 \) atm (since we're dealing with normal atmospheric pressure).

### Steps:
1. **Calculate the volume of the spherical air tank:**
   The formula for the volume of a sphere is:
   \[
   V = \frac{4}{3} \pi r^3
   \]
   where \( r \) is the radius of the tank. Given the diameter of 89.0 cm, the radius is:
   \[
   r = \frac{89.0}{2} = 44.5 \, \text{cm}
   \]
   Convert this to meters for consistency in units:
   \[
   r = 0.445 \, \text{m}
   \]

2. **Convert tank volume from cubic meters to liters:**
   1 cubic meter (m³) is equal to 1000 liters (L).

3. **Apply Boyle's Law:**
   \[
   P_1 V_1 = P_2 V_2
   \]
   where \( P_1 = 1 \) atm, \( V_1 = 9000 \) L, and \( V_2 \) is the volume of the tank (in liters).

Let me calculate these steps for you.The spherical air tank has a volume of approximately **369.12 liters**. To compress 9000 liters of air into this tank, the required pressure would be about **24.38 atmospheres (atm)**.

Would you like further details or have any questions?

Here are some related questions to explore further:
1. How does Boyle's Law apply to other types of containers, like cylinders?
2. What other variables could affect the pressure needed in real-world scenarios?
3. How would the pressure change if the tank had a different shape, like a cube or cylinder?
4. What safety precautions must be considered when compressing gases to high pressures?
5. How do modern submersibles manage air pressure to ensure the safety of divers?

**Tip:** Always double-check unit conversions, especially when dealing with volumes in liters and cubic meters!

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 89.0 cm wide. The biologist estimates she will need 9000 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 air pressure needed to compress 9000 L of air into a spherical tank using Boyle's Law. This problem includes finding the volume of a spherical tank with a diameter of 89.0 cm and applying Boyle's Law to determine the pressure in atmospheres.

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