(a) Calculate the absolute pressure at the bottom of a fresh-water lake at a depth of 25.6 m. Assume the density of the water is 1.00 103 kg/m3 and the air above is at a pressure of 101.3 kPa.

(b) What force is exerted by the water on the window of an underwater vehicle at this depth if the window is circular and has a diameter of 33.1 cm?

To calculate the absolute pressure at the bottom of a freshwater lake, we can use the hydrostatic pressure formula:

P = P₀ + ρgh

where:
P is the absolute pressure at the bottom,
P₀ is the pressure at the surface (101.3 kPa),
ρ is the density of water (1.00 x 10³ kg/m³),
g is the acceleration due to gravity (approximately 9.8 m/s²),
h is the depth of the lake (25.6 m).

(a) Let's substitute the given values into the formula to find the answer:

P = 101.3 kPa + (1.00 x 10³ kg/m³)(9.8 m/s²)(25.6 m)
P = 101.3 kPa + 250.88 kPa
P = 352.18 kPa

Therefore, the absolute pressure at the bottom of the lake is 352.18 kPa.

To calculate the force exerted by the water on the window of an underwater vehicle at this depth, we can use the formula:

F = P * A

where:
F is the force exerted by the water,
P is the absolute pressure at the depth (352.18 kPa),
A is the cross-sectional area of the window.

(b) First, let's calculate the area of the window:

A = π * r²

where:
A is the area,
π is a mathematical constant approximately equal to 3.14,
r is the radius of the window (half of the diameter).

Given that the diameter of the window is 33.1 cm, we can convert it to meters and calculate the radius:

r = 33.1 cm / 100 cm/m / 2
r = 0.1655 m

Now we can substitute the values into the formula to find the force:

F = (352.18 kPa)(π)(0.1655 m)²

Calculating the expression will give you the force exerted by the water on the window in units of force (newtons).