Question

if pressure at half the depth of a lake is equal to 2/3 pressure at the bottom of the lake then what is the depth of the lake ?

(a) 10m
(b) 20m
(c) 60m
(d) 30m

Answer 1

Depth = d
Atmospheric Pressure = P₀ = 10⁵ Pa
Density of water at 4°C (ρ) = 10³ kg/m³

Pressure at half of depth
P = P₀ + (d/2)ρg
P = P₀ + 0.5dρg

Pressure at the bottom of lake
P’ = P₀ + dρg

Given that P = 2P’/3
P₀ + 0.5dρg = 2(P₀ + dρg)/3
P₀ + 0.5dg = 2P₀/3 + 2dρg/3
P₀/3 = dρg/6
d = 2P₀ / (ρg)
d = (2 × 10⁵ Pa) / (10³ kg/m³ × 10 m/s²)
d = 20 m

Depth of lake is 20 m

Answer 2

Depth = d

Atmospheric Pressure = P₀ = 10⁵ Pa

Density of water at 4°C (ρ) = 10³ kg/m³

Pressure at half of depth

P = P₀ + (d/2)ρg

P = P₀ + 0.5dρg

Pressure at the bottom of lake

P’ = P₀ + dρg

Given that P = 2P’/3

P₀ + 0.5dρg = 2(P₀ + dρg)/3

P₀ + 0.5dg = 2P₀/3 + 2dρg/3

P₀/3 = dρg/6

d = 2P₀ / (ρg)

d = (2 × 10⁵ Pa) / (10³ kg/m³ × 10 m/s²)

d = 20 m