To find the height of the hill based on the information provided about the gas balloon, we can use the ideal gas law and the relationship between pressure and volume. Let's break this down step by step.
Understanding the Problem
We know that at sea level, the atmospheric pressure (Patm) is 1 bar. The volume of the gas balloon increases to 6/5 times its original volume when it reaches the top of the hill. The pressure at height h is given by the equation:
P = Patm(1 - h/80)
Here, Patm is the atmospheric pressure at sea level, and h is the height in meters. Our goal is to find the height h of the hill.
Applying the Ideal Gas Law
The ideal gas law states that for a given amount of gas at constant temperature, the pressure and volume are inversely related. This can be expressed as:
P1 * V1 = P2 * V2
Where:
- P1 = initial pressure at sea level (1 bar)
- V1 = initial volume of the balloon
- P2 = pressure at height h
- V2 = final volume of the balloon (6/5 V1)
Setting Up the Equation
Substituting the known values into the ideal gas law gives us:
1 bar * V1 = P2 * (6/5 V1)
We can simplify this equation by canceling out V1 (assuming it is not zero):
1 bar = P2 * (6/5)
From this, we can solve for P2:
P2 = (5/6) bar
Finding the Height of the Hill
Now we can use the pressure equation provided to find the height h:
P2 = Patm(1 - h/80)
Substituting Patm = 1 bar and P2 = (5/6) bar into this equation gives:
(5/6) = 1(1 - h/80)
Now, we can solve for h:
(5/6) = 1 - h/80
h/80 = 1 - (5/6)
h/80 = 1/6
Multiplying both sides by 80 gives:
h = (80/6)
h = 13.33 meters
Final Result
The height of the hill is approximately 13.33 meters. This calculation illustrates how changes in pressure and volume of a gas can be used to determine altitude, demonstrating the principles of gas behavior in different atmospheric conditions.
