The water in an 8-m-diameter, 3-m-high above-ground swimming pool is to be emptied by unplugging a 6-cm-diameter, 25-m-long horizontal pipe attached to the bottom of the pool. Determine how long it will take to empty the swimming pool completely.

 

The draining time of the swimming pool is h.

To determine how long it will take to empty the swimming pool, we can approach the problem using Torricelli’s law, which relates the velocity of the water exiting the pipe to the height of the water in the pool. The overall process involves these steps:

1. **Given information:**
   - Diameter of the pool: \(D_{\text{pool}} = 8 \, \text{m}\)
   - Height of the pool: \(h_{\text{pool}} = 3 \, \text{m}\)
   - Diameter of the pipe: \(D_{\text{pipe}} = 6 \, \text{cm} = 0.06 \, \text{m}\)
   - Length of the pipe: \(L_{\text{pipe}} = 25 \, \text{m}\)

2. **Step 1: Calculate the volume of the swimming pool**  
   The swimming pool is cylindrical, and its volume is given by:
   \[
   V_{\text{pool}} = \pi \left(\frac{D_{\text{pool}}}{2}\right)^2 h_{\text{pool}}
   \]
   Substituting the values:
   \[
   V_{\text{pool}} = \pi \left(\frac{8}{2}\right)^2 \times 3 = \pi \times 4^2 \times 3 = 48\pi \, \text{m}^3
   \]

3. **Step 2: Apply Torricelli’s law for draining velocity**  
   Torricelli's law states that the velocity of water at the bottom of the pool is:
   \[
   v = \sqrt{2gh}
   \]
   where:
   - \(g = 9.81 \, \text{m/s}^2\) is the acceleration due to gravity,
   - \(h\) is the height of water above the drain at any time \(t\).

   As the pool drains, the height \(h(t)\) decreases, so the draining rate decreases over time.

4. **Step 3: Calculate the flow rate**  
   The flow rate \(Q(t)\) is the product of the velocity and the cross-sectional area of the pipe:
   \[
   Q(t) = A_{\text{pipe}} v(t) = \left(\frac{\pi D_{\text{pipe}}^2}{4}\right) \sqrt{2g h(t)}
   \]
   Substituting the known values of the pipe's diameter and simplifying:
   \[
   Q(t) = \frac{\pi (0.06)^2}{4} \sqrt{2 \times 9.81 \times h(t)} = \frac{\pi \times 0.0036}{4} \sqrt{19.62 h(t)}
   \]

5. **Step 4: Relate the flow rate to the rate of change of water height**  
   The rate of change of the water volume in the pool is related to the height of the water as follows:
   \[
   \frac{dV_{\text{pool}}}{dt} = -Q(t)
   \]
   The volume of the pool is related to the height by:
   \[
   V_{\text{pool}} = A_{\text{pool}} h(t) = \pi \left(\frac{D_{\text{pool}}}{2}\right)^2 h(t) = 16\pi h(t)
   \]
   So the rate of change of height is:
   \[
   A_{\text{pool}} \frac{dh}{dt} = -Q(t)
   \]
   Substituting for \(Q(t)\) from step 3:
   \[
   16\pi \frac{dh}{dt} = -\frac{\pi \times 0.0036}{4} \sqrt{19.62 h(t)}
   \]

6. **Step 5: Solve the differential equation to find the time**  
   Separating variables and integrating will give the total time to drain the pool completely:
   \[
   \int_0^{h_0} \frac{dh}{\sqrt{h}} = -\frac{0.0036}{4 \times 16} \sqrt{19.62} \int_0^t dt
   \]

Let me now compute the integral and solve for the total draining time \(t\).The time required to completely empty the swimming pool is approximately 272,783 seconds, or around 75.8 hours.

Would you like more details or have any questions?

Here are some related questions for further understanding:
1. How would the draining time change if the pipe's diameter were increased?
2. What if the pipe were shorter—how would that impact the draining time?
3. How does the height of the pool affect the draining time according to Torricelli's law?
4. Can the shape of the pool (e.g., conical or rectangular) affect the draining time?
5. How does water pressure change as the pool drains?

**Tip:** The draining time depends significantly on the diameter of the pipe since it affects the flow rate. Increasing the pipe's diameter speeds up draining time.

The water in an 8-m-diameter, 3-m-high above-ground swimming pool is to be emptied by unplugging a 6-cm-diameter, 25-m-long horizontal pipe attached to the bottom of the pool. Determine how long it will take to empty the swimming pool completely.

Learn how to calculate the time required to empty an 8-meter diameter swimming pool using a 6-cm diameter pipe based on Torricelli’s Law. The solution involves fluid dynamics, volume calculations, and solving a differential equation. Suitable for high school students in Grades 11-12.

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