Question

When an air bubble rises from the bottom to the top of a freshwater lake, its volume increases by 80%. If the temperatures at the bottom and the top of the lake are 4.0 and 10 °C, respectively, how deep is the lake?

a) 8.0 m

b) 10.0 m

c) 12.0 m

d) 14.0 m

Answer 1

When an air bubble rises from the bottom of a freshwater lake to the top, its volume increases by 80%. Using Charles's Law, we can relate the temperature and volume of the bubble.

By solving for the depth of the lake using the relationship between pressure and depth in a fluid, we find that the depth of the lake is approximately 3.57m, closest to option a) 3.6m.

Step-by-step explanation:

When an air bubble rises from the bottom to the top of a freshwater lake, its volume increases by 80%. In this scenario, we can use Charles's Law to relate the temperature and volume of the bubble. Charles's Law states that the volume of a gas is directly proportional to its temperature, assuming pressure is constant. So, as the bubble rises and the temperature increases from 4.0 to 10 °C, the volume of the bubble will increase by 80%. This means that the temperature change of the bubble from bottom to top is directly proportional to the volume change:

(10 - 4) °C / 4 °C = 80% / 100% = 0.8

Now we can solve for the depth of the lake. Let's assume the depth of the lake is x meters. We can use the relationship between pressure and depth in a fluid to solve for x.

Pressure is directly proportional to depth:

(10m - x) / x = 1.8

Cross-multiplying and simplifying, we get:

10m - x = 1.8x

Combining like terms:

2.8x = 10m

Dividing by 2.8, we find that x = 3.57m.

Therefore, the depth of the lake is approximately 3.57m, so the closest option to this is a) 3.6 m.