A scuba diver exhales,releasing a spherical bubble of gas with a diameter of 10 cm.The bubble floats towards the surface maintaining its spherical shape.When the bubble is 1.0 m below the surface,its diameter is 16 cm.How far below the water's surface is the diver?

Question

Quizplus · Accepted Answer

We can use the equation PV = k, where P is pressure, V is volume and k is a constant. Since the bubble floats towards the surface, the pressure inside the bubble is equal to the pressure outside the bubble when it is at the surface. Let's call this pressure P0. When the bubble is 1.0 m below the surface, the pressure inside the bubble is equal to the pressure at that depth, let's call this pressure P1. Since the bubble maintains its spherical shape, we can use the equation for the volume of a sphere:
V = (4/3)πr^3

When the bubble is at the surface, its diameter is 10 cm, so its radius is 5 cm. Therefore, its volume is:
V0 = (4/3)π(5 cm)^3 = 523.6 cm^3

When the bubble is 1.0 m below the surface, its diameter is 16 cm, so its radius is 8 cm. Therefore, its volume is:
V1 = (4/3)π(8 cm)^3 = 2144 cm^3

Since PV is constant, we have:
P0V0 = P1V1

We can solve for P0:
P0 = P1V1/V0 = (1 atm)(2144 cm^3)/(523.6 cm^3) = 4.08 atm

We can use the equation for pressure in a fluid:
P = ρgh

Where ρ is the density of water (1000 kg/m^3), g is the acceleration due to gravity (9.81 m/s^2), and h is the depth below the surface. Solving for h, we get:
h = P/(ρg) = (4.08 atm)(101325 Pa/atm)/(1000 kg/m^3)(9.81 m/s^2) = 36.9 m

Therefore, the diver is 36.9 m below the water's surface. 
The best choice is B, 36 m.

A Scuba Diver Exhales,releasing a Spherical Bubble of Gas with a Diameter