A 5/8-in. (inside) diameter garden hose is used to fill a round swimming pool 4.6 m in diameter. Part A How long will it take to fill the pool to a depth of 1.6 m if water issues from the hose at a speed of 0.59 m/s?

t = 2.26 10⁵ s

Explanation:

Let's calculate the flow of water that comes out of the hose

Q = A v

The area of a circle is

A = π r²

Q = π r² v

Let's reduce the units to the SI system

d = 5/8 in (2.54 10⁻² m / in) = 1.5875 10⁻² m

r = d / 2 = 0.79375 10⁻² m

Q = π (0.79375 10⁻²) ² 0.59

Q = 1.1678 10⁻⁴ m³ / s

Let's calculate the volume of the pool

V = π R² h

V = π (4.6 / 2) ² 1.6

V = 26.59 m³

Let's use a rule of proportions (rule of three), to find the time

t = 26.59 (1 / 1.1678 10⁻⁴)

t = 22.77 10⁴ s

t = 2.26 10⁵ s

Time taken to fill the pool (t) = 223529.41 s

Explanation:

The volume of water in the pool (V) = πd²h/4 ... equation 1

Where d = radius of the pool, h = height of the pool

At a height of 1.6 m, the volume of water is

V (pool) = (3.143 * 4.6² * 1.6) / 4 = 26.60 m³.

The volume of the hose is

V (hose) = πd²h/4 ... equation 2

And the rate of flow of water from the hose into the pool is

V (hose) / dt = (dh/dt) * (dV (hose) / dh)

Differentiating equation 2,

V (hose) / dt = πd²/4

where d = 5/8-in. = (5/8) * 0.0254 = 0.016 m, π = 3.143, dh/dt = 0.59 m/s

dV (hose) / dt = { (3.143*0.016²) / 4} * 0.59

dV (hose) / dt = 0.000119 m³/s.

∴ time taken to fill the pool (t) = V (pool) / (dV (hose) / dt)

t = 26.60/0.000119

t = 223529.41 s.

Time taken to fill the pool (t) = 223529.41 s