MCAT contradiction? - Gauge Pressure

[berkeley12]

Homework Statement

A large cylinder is filled with an equal volume of two immiscible fluids. A balloon is submerged in the first fluid; the gauge pressure in the balloon at the deepest point in the first fluid is found to be 3 atm. Next, the balloon is lowered all the way to the bottom of the cylinder, and as it is submerged in the second fluid, the hydrostatic pressure in the balloon reads 8 atm. What is the gauge pressure at the bottom of the second fluid?

Homework Equations

according to the book:
P_gauge = P_absolute - P_atm
P_absolute = P_0 + $$\rho$$gh

The Attempt at a Solution

The hydrostatic pressure is the same as absolute pressure, so the hydrostatic pressure of the bottom fluid 8 atm.

P_gauge for the bottom liquid is: 8 - (P_atm) according to the formula given. So that's 8 - 1= 7atm.
But in the MCAT book, the P_gauge for the bottom is 4, not 7.

The book does: 8 - (1 + 3), where that 3 comes from the gauge pressure from the upper liquid. I can see this making sense. Basically I just want to know, is the formula given for gauge pressure not necessarily accurate? I feel like the "P_atm" in the formula isn't accurate, since in this problem, using it didn't work. So P_atm can adjust if there's additional pressure above the liquid? Thank you.

 

[ideasrule]

What is the ratio of the gauge pressure at the bottom of the second fluid?

The ratio between this and what?

But in the MCAT book, the P_gauge for the bottom is 4, not 7.

The book is wrong.

The book does: 8 - (1 + 3), where that 3 comes from the gauge pressure from the upper liquid.

That's the pressure exerted by the bottom fluid, not the gauge pressure at the bottom of the second fluid. Is that what the question is asking for?

Basically I just want to know, is the formula given for gauge pressure not necessarily accurate?

It's always accurate. P_atm is just the pressure of the atmosphere. Earth's atmosphere doesn't care what your beaker looks like or what fluids it contains; it always pushes down with the same pressure.

 

[berkeley12]

Whoops. Basically, they want the gauge pressure of the bottom liquid.

In the answer to the problem, when calculating the gauge pressure of the bottom liquid, they also subtract off the 3atm pressure from the upper liquid. The book's answer is 5, which comes from: 8 - (1 +3).

You're telling me that I should be doing: 8 - 1. Doesn't it make sense to factor in the pressure from the upper liquid in somewhere though?

 

[ideasrule]

Whoops. Basically, they want the gauge pressure of the bottom liquid.

In that case, 8-(1+3) is correct. 8-1 is the gauge pressure at the bottom of the bottom fluid, but the pressure contribution from the bottom fluid itself is only 4 atm.

 

[berkeley12]

Perfect, thanks for your help.