Understanding the Water Flow Problem

This problem involves calculating how much the water level in a cylindrical tank will increase over a specific time period, given the dimensions of the circular pipe through which water flows and the speed of the water. We need to find the rise in water level.

Gathering Key Information

Let's list the given values:

• Internal diameter of the circular pipe = 2 cm
• Rate (speed) of water flow = 4 metres per second
• Radius of the base of the cylindrical tank = 80 cm
• Time duration = 16 minutes

Step-by-Step Calculation of Water Level Rise

1. Calculate the Pipe's Radius and Area

The radius of the pipe is half its diameter. We also need to convert the flow rate to consistent units (cm/s).

• Pipe radius ($r_{pipe}$): $ \frac{\text{Diameter}}{2} = \frac{2 \text{ cm}}{2} = 1 \text{ cm} $
• Flow speed ($v$): $ 4 \text{ m/s} = 4 \times 100 \text{ cm/s} = 400 \text{ cm/s} $
• Cross-sectional area of the pipe ($A_{pipe}$): $ A_{pipe} = \pi \times (r_{pipe})^2 = \pi \times (1 \text{ cm})^2 = \pi \text{ cm}^2 $

2. Calculate the Volume of Water Flowing Per Second

The volume of water flowing per second is the cross-sectional area of the pipe multiplied by the speed of the water.

• Volume flow rate ($V_{rate}$): $ V_{rate} = A_{pipe} \times v = (\pi \text{ cm}^2) \times (400 \text{ cm/s}) = 400\pi \text{ cm}^3/\text{s} $

3. Calculate the Total Time in Seconds

The time is given in minutes, so we convert it to seconds for consistency.

• Total time ($t$): $ 16 \text{ minutes} = 16 \times 60 \text{ seconds} = 960 \text{ seconds} $

4. Calculate the Total Volume of Water Flowed

Multiply the volume flow rate by the total time to find the total volume of water that enters the tank.

• Total Volume ($V_{total}$): $ V_{total} = V_{rate} \times t = (400\pi \text{ cm}^3/\text{s}) \times (960 \text{ s}) $
• $ V_{total} = 384000\pi \text{ cm}^3 $

5. Calculate the Base Area of the Cylindrical Tank

The tank is cylindrical, and we are given its base radius.

• Tank base radius ($R_{tank}$): $ 80 \text{ cm} $
• Area of the tank's base ($A_{tank}$): $ A_{tank} = \pi \times (R_{tank})^2 = \pi \times (80 \text{ cm})^2 $
• $ A_{tank} = \pi \times 6400 \text{ cm}^2 = 6400\pi \text{ cm}^2 $

6. Calculate the Rise in Water Level

The total volume of water in the tank divided by the base area of the tank gives the height (rise) of the water level.

• Rise in water level ($h$): $ h = \frac{V_{total}}{A_{tank}} = \frac{384000\pi \text{ cm}^3}{6400\pi \text{ cm}^2} $
• $ h = \frac{384000}{6400} \text{ cm} $
• $ h = \frac{3840}{64} \text{ cm} $
• $ h = 60 \text{ cm} $

Final Answer Summary

After performing the calculations step-by-step, considering the pipe dimensions, the flow rate, and the cylindrical tank's base area, the total volume of water that flows into the tank in 16 minutes results in a water level rise of 60 cm.