Question

a cubical box 25.0 cm on each side is immersed in a fluid. the pressure at the top surface of the box is 109.40 kpa and the pressure on the bottom surface is 112.00 kpa. what is the density of the fluid?
a) 1060 kg/m3
b) 1000 kg/m3
c) 1030 kg/m3
d) 1090 kg/m3
e) 1120 kg/m3

Explanation:

Step1: Recall hydrostatic pressure formula

The hydrostatic pressure difference between two points in a fluid is given by \(\Delta P =
ho g h\), where \(\Delta P\) is the pressure difference, \(
ho\) is the density of the fluid, \(g\) is the acceleration due to gravity (\(9.8\ m/s^2\)), and \(h\) is the vertical distance between the two points.

Step2: Calculate pressure difference

First, find the pressure difference \(\Delta P\). The pressure at the bottom is \(P_{bottom}=112.00\ kPa = 112000\ Pa\) and at the top is \(P_{top}=109.40\ kPa = 109400\ Pa\). So \(\Delta P=P_{bottom}-P_{top}=112000 - 109400=2600\ Pa\).

Step3: Determine the height \(h\)

The cubical box has sides of length \(25.0\ cm = 0.25\ m\). The vertical distance between the top and bottom surfaces of the box (which is the height \(h\) in the hydrostatic pressure formula) is equal to the side length of the cube, so \(h = 0.25\ m\).

Step4: Solve for density \(

ho\)
From \(\Delta P=
ho g h\), we can rearrange to solve for \(
ho\): \(
ho=\frac{\Delta P}{gh}\). Substitute \(\Delta P = 2600\ Pa\), \(g = 9.8\ m/s^2\), and \(h = 0.25\ m\) into the formula: \(
ho=\frac{2600}{9.8\times0.25}=\frac{2600}{2.45}\approx1061.22\ kg/m^3\), which is approximately \(1060\ kg/m^3\).

Answer:

a) 1060 kg/m³