To find the radius of the bubble when it reaches the surface, we can use Boyle's Law, which states that the pressure of a gas times its volume is constant if temperature is held constant. Mathematically, this can be expressed as:
\[ P_1 V_1 = P_2 V_2 \]
Where:
- \( P_1 \) = initial pressure
- \( V_1 \) = initial volume
- \( P_2 \) = final pressure
- \( V_2 \) = final volume
Since the volume of a sphere is given by \( V = \frac{4}{3} \pi r^3 \), we can relate the volumes in terms of radius:
\[ V = \frac{4}{3} \pi r^3 \]
Thus, we can express Boyle's Law in terms of the radii:
\[ P_1 \left(\frac{4}{3} \pi r_1^3\right) = P_2 \left(\frac{4}{3} \pi r_2^3\right) \]
The \( \frac{4}{3} \pi \) cancels out, simplifying to:
\[ P_1 r_1^3 = P_2 r_2^3 \]
Now, we can plug in the values.
Given:
- \( r_1 = 2.9 \) cm
- \( P_1 = 2.16 \) atm (at depth)
- \( P_2 = 1.01 \) bar (atmospheric pressure at the surface)
First, we convert \( P_1 \) from atm to bars for consistency since \( 1 \) atm \( \approx 1.013 \) bar. Thus:
\[ P_1 = 2.16 \text{ atm} \times 1.013 \text{ bar/atm} \approx 2.19 \text{ bar} \]
Now, substituting into Boyle's Law:
\[ 2.19 \times (2.9)^3 = 1.01 \times r_2^3 \]
Calculating \( (2.9)^3 \):
\[ (2.9)^3 \approx 24.389 \]
Now substituting back into the equation:
\[ 2.19 \times 24.389 = 1.01 \times r_2^3 \]
Calculating \( 2.19 \times 24.389 \):
\[ 2.19 \times 24.389 \approx 53.465 \]
Now we have:
\[ 53.465 = 1.01 \times r_2^3 \]
To find \( r_2^3 \):
\[ r_2^3 = \frac{53.465}{1.01} \approx 52.017 \]
Now, taking the cube root to find \( r_2 \):
\[ r_2 \approx \sqrt[3]{52.017} \approx 3.73 \text{ cm} \]
Finally, rounding to two significant figures, the radius of the bubble at the surface is:
Answer: \( r_2 \approx 3.7 \) cm.
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