Question

(II)A\(\frac{5}{8}\;{\rm{in}}\). (inside) diameter garden hose is used to fill a round swimming pool 6.1 m in diameter. How long will it take to fill the pool to a depth of 1.4 m if water flows from the hose at a speed of 0.40 m/s?

Short answer

The time taken to fill the pool is \(5.88\;{\rm{days}}\).

Step-by-step solution

Understanding the of volume flow rate

In this problem, the volume flow rate can be calculated by using the product of area as well as velocity of fluid.

Given data

The diameter of garden hose is \(d = \frac{5}{8}\;{\rm{in}}\).

The diameter of the round swimming pool is \(D = 6.1\;{\rm{m}}\).

The depth of the pool is \(h = 1.4\;{\rm{m}}\).

The speed of water flows from the hose is \(v = 0.40\;{\rm{m/s}}\).

Evaluating the volume of swimming pool and area of hose

The volume of swimming pool is calculated as:

\(V = \frac{\pi }{4}{D^2}h\)

Substitute the values in the above equation.

\(\begin{array}{c} = \frac{\pi }{4}{\left( {6.1\;{\rm{m}}} \right)^2}\left( {1.4\;{\rm{m}}} \right)\\ = 40.91\;{{\rm{m}}^3}\end{array}\)

The area of hose is calculated below:

\(A = \frac{\pi }{4}{d^2}\)

Substitute the values in the above equation.

\(\begin{array}{c} = \frac{\pi }{4}{\left( {\frac{5}{8}\;{\rm{in}} \times \frac{{0.0254\;{\rm{m}}}}{{1\;{\rm{in}}}}} \right)^2}\\ = 2.01 \times {10^{ - 4}}\;{{\rm{m}}^2}\end{array}\)

Evaluating the time taken by hose to fill the pool

The time taken by hose to fill the pool is calculated below:

\(\begin{array}{c}\frac{V}{{\Delta t}} = A \cdot v\\\Delta t = \frac{V}{{A \cdot v}}\end{array}\)

Substitute the values in the above equation.

\(\begin{array}{c} = \frac{{40.91\;{{\rm{m}}^3}}}{{\left( {2.01 \times {{10}^{ - 4}}\;{{\rm{m}}^2}} \right)\left( {0.40\;{\rm{m/s}}} \right)}}\\ = 508674.1\;{\rm{s}} \times \left( {\frac{{1\;{\rm{day}}}}{{86400\;{\rm{s}}}}} \right)\\ = 5.88\;{\rm{days}}\end{array}\)

Hence, the time taken by hose to fill the pool is \(5.88\;{\rm{days}}\).