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To solve this problem, we need to calculate both the absolute and gauge pressures at point \( A \). Here’s a step-by-step breakdown of the process:

### Given:
- We are working with a water-filled system, which means the fluid density \(\rho_{\text{water}}\) is \(1000 \, \text{kg/m}^3\).
- Barometric pressure: 1 atmosphere (\( P_{\text{atm}} = 1.0 \, \text{atm} = 1.013 \times 10^5 \, \text{Pa} \)).
- Point \( A \) is located at a height of 76 cm from the bottom (or the water surface) in the tank.
- The mercury column has a height of 13.6 cm (or 0.136 m) with a specific gravity (\( SG_{\text{Hg}} \)) of 13.6.

### Step 1: Gauge Pressure at Point \( A \)

Gauge pressure is the pressure exerted by the liquid column above point \( A \). Using the hydrostatic pressure formula:

\[
P_{\text{gauge}} = \rho g h
\]

Where:
- \(\rho\) is the density of the fluid (water in this case: \(1000 \, \text{kg/m}^3\)),
- \(g\) is the acceleration due to gravity (\(9.81 \, \text{m/s}^2\)),
- \(h\) is the height of the water column (76 cm = 0.76 m).

\[
P_{\text{gauge}} = 1000 \, \text{kg/m}^3 \times 9.81 \, \text{m/s}^2 \times 0.76 \, \text{m}
\]

Let me calculate this value.The gauge pressure at point \( A \) due to the water column is approximately \( 7.46 \, \text{kPa} \).

### Step 2: Absolute Pressure at Point \( A \)

The absolute pressure is the sum of the gauge pressure and the atmospheric pressure:

\[
P_{\text{absolute}} = P_{\text{gauge}} + P_{\text{atm}}
\]

Given that \( P_{\text{atm}} = 101.3 \, \text{kPa} \):

\[
P_{\text{absolute}} = 7.46 \, \text{kPa} + 101.3 \, \text{kPa}
\]

Let me calculate this value.It seems like I can't do more advanced data analysis right now. Please try again later. 

However, you can calculate the absolute pressure using the formula:

\[
P_{\text{absolute}} = P_{\text{gauge}} + P_{\text{atm}} = 7.46 \, \text{kPa} + 101.3 \, \text{kPa} = 108.76 \, \text{kPa}
\]

If you need further clarification or assistance with these steps, feel free to ask!

### Would you like more details or have any questions? Here are some related questions:

1. How is the barometric pressure typically measured in real applications?
2. Why is it important to differentiate between absolute and gauge pressure?
3. What happens to the gauge pressure if the liquid is replaced with a denser one, like mercury?
4. How does temperature affect pressure calculations in a fluid system?
5. What are some practical applications of calculating hydrostatic pressure?

**Tip:** When working with pressure systems, always ensure units are consistent to avoid calculation errors.

Calcular la presión absoluta y manométrica en el punto A en kg/cm^2. Considerar que la presión barométrica es de 1.0 atmósferas.

This guide provides a step-by-step solution to calculating both the absolute and gauge pressures in a water-filled system at point A. Includes key formulas such as P_gauge = ρgh and P_absolute = P_gauge + P_atm. Suitable for university-level students in fluid mechanics.

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