Question

The ratio of the diameters of certain air bubble at the bottom and at the surface is $1:2.$ What is the depth of the lake? $($ 1 atmosphere $=10\text{m}$ depth of water$)$

• A. $80\text{m}$
• B. $60\text{m}$
• C. $35\text{m}$
• D. $70\text{m}$

Answer 1

The correct option is D $70\text{m}$

Lets say depth of the lake is $H\text{m}$, and we have $r_2=2r_1$
$P_1{V}_1=P_2{V}_2$
$\Rightarrow (P_o+H\rho g)\frac{4}{3}\pi r_1^3=P_o\frac{4}{3}\pi r_2^3$
$\Rightarrow (P_o+H\rho g)r_1^3=P_o(2r_1{)}^3$
$\Rightarrow P_o+H\rho g=8P_o$
$\Rightarrow P_o=\frac{H}{7}\rho g$
$\Rightarrow \frac{H}{7}=10m\because (\text{1 atmosphere = 10 m depth of water)}$
$\Rightarrow H=70\text{m}$