Pressure question

[arc5005]

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At a specific depth in a swimming pool, a barometer measures the total pressure to be twice that of atmospheric pressure. If the barometer is now submerged to a depth that is twice its initial depth, by how much does the total pressure increase?

A. 50%
B. 100%
C. 200%
D. 300%

A) 50%

We can use the ratio technique to handle this percentage change type of problem. The relevant situation here is the relationship of pressure to depth:

Ptotal = Patm + ρgh

Applying this to the first depth implies that: ρghinitial = Patm, since the total pressure there is 2Patm. If the second depth is twice that of the first, then ρghfinal =2Patm. This means that the total pressure at the second depth is 3Patm. Taking the ratio of the final to the initial situation gives:

Ptotal final / Ptotal initial = 3Patm / 2Patm = 1.5

To complete this problem, just subtract 1 from the ratio to get the percentage change of +0.5 (i.e. 50%). Be careful of the wording in percentage questions. Here, we want a "percentage change." If the question had asked instead, "what percentage of the initial total pressure is the final total pressure?", the answer would have been: "The final total pressure is 150% of the initial total pressure."

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This is confusing to me. I don't think I understand the ratio concept here, and why they subtract 1 from the ratio to get a percentage change of +.5

 

[NextStepTutor_3]

Hi @arc5005! Great question.

As with most MCAT questions, there are multiple ways to answer this question. If the ratio method doesn't make sense to you, then rather than spending a lot of time trying to figure it out, we can simply find another method. Here, the reason why they use the ratio method is because no numerical values were given. However, we can easily obtain the numerical values we need and then use them, as follows:

First, let's assume the atmospheric pressure is standard: 1 atm. (The value/units we choose here doesn't matter as long as we stay consistent, but 1 atm is very easy to work with.)

This means that when the "barometer measures the total pressure to be twice that of atmospheric pressure," it must be measuring a total pressure of 2 atm. The gauge pressure - or the pressure not including atmospheric pressure - must be 1 atm here.

Next, the barometer is "submerged to a depth that is twice its initial depth." Depth and gauge pressure are proportional, according to the equation Pgauge = ρgh. So, doubling the depth should double the gauge pressure from 1 atm to 2 atm. But the question didn't ask about gauge pressure - it asked about total pressure, so we need to factor the atmospheric pressure back in. This gives us a final pressure of 3 atm.

To get our final answer, simply ask yourself what percent change must take place to increase from 2 atm to 3 atm. The answer is a 50% increase (since we increased by 1, and 1 is 50% of 2).



Now, even though we didn't use the ratio method, I can still answer your second question here, as it's an important concept. Take a look at the part of the explanation that says "Ptotal final / Ptotal initial = 3Patm / 2Patm = 1.5." This means that Ptotal final / Ptotal initial = 1.5, or that the final total pressure is 1.5 times the initial total pressure. (We now know this is true, since the final total pressure is 3, while the initial total pressure was only 2.) Since the question asks for a percentage, we need to convert "1.5 times" to "150 percent."

Now, the key is that 150% is not our answer. 150% just means that 3 is 150% of 2. But this question didn't ask about 3 (the final total pressure). It asked about the increase from 2 to 3. We thus need to take 150% and subtract 100% to get 50%. We do this because the 2 was the pressure that was already there, and 2 is 100% of itself. So taking 150% - 100% is similar to taking 3 atm - 2 atm, which gives us the difference between them that we're looking for.

Let me know if I can clarify anything further or help with anything else 🙂

 

[arc5005]

Hi @arc5005! Great question.

As with most MCAT questions, there are multiple ways to answer this question. If the ratio method doesn't make sense to you, then rather than spending a lot of time trying to figure it out, we can simply find another method. Here, the reason why they use the ratio method is because no numerical values were given. However, we can easily obtain the numerical values we need and then use them, as follows:

First, let's assume the atmospheric pressure is standard: 1 atm. (The value/units we choose here doesn't matter as long as we stay consistent, but 1 atm is very easy to work with.)

This means that when the "barometer measures the total pressure to be twice that of atmospheric pressure," it must be measuring a total pressure of 2 atm. The gauge pressure - or the pressure not including atmospheric pressure - must be 1 atm here.

Next, the barometer is "submerged to a depth that is twice its initial depth." Depth and gauge pressure are proportional, according to the equation Pgauge = ρgh. So, doubling the depth should double the gauge pressure from 1 atm to 2 atm. But the question didn't ask about gauge pressure - it asked about total pressure, so we need to factor the atmospheric pressure back in. This gives us a final pressure of 3 atm.

To get our final answer, simply ask yourself what percent change must take place to increase from 2 atm to 3 atm. The answer is a 50% increase (since we increased by 1, and 1 is 50% of 2).



Now, even though we didn't use the ratio method, I can still answer your second question here, as it's an important concept. Take a look at the part of the explanation that says "Ptotal final / Ptotal initial = 3Patm / 2Patm = 1.5." This means that Ptotal final / Ptotal initial = 1.5, or that the final total pressure is 1.5 times the initial total pressure. (We now know this is true, since the final total pressure is 3, while the initial total pressure was only 2.) Since the question asks for a percentage, we need to convert "1.5 times" to "150 percent."

Now, the key is that 150% is not our answer. 150% just means that 3 is 150% of 2. But this question didn't ask about 3 (the final total pressure). It asked about the increase from 2 to 3. We thus need to take 150% and subtract 100% to get 50%. We do this because the 2 was the pressure that was already there, and 2 is 100% of itself. So taking 150% - 100% is similar to taking 3 atm - 2 atm, which gives us the difference between them that we're looking for.

Let me know if I can clarify anything further or help with anything else 🙂

Thank you I appreciate this!