Water flows through a pipe (inner radius 1 cm) at 80 cm/s into an empty cylindrical tank of radius 40 cm. What is the rise in water level in half an hour?
90 cm
Step 1: Write down given data.
Step 2: Find volume of water flowing into the tank in 1800 s.
Volume flow per second through pipe = Cross-sectional area × velocity
\( A_{pipe} = \pi r^2 = \pi (0.01)^2 = \pi (0.0001) = 0.0001\pi \, m^2 \)
Flow per second = \( A_{pipe} \times v = 0.0001\pi \times 0.8 = 0.00008\pi \, m^3/s \)
Total volume in 1800 s = \( 0.00008\pi \times 1800 = 0.144\pi \, m^3 \)
Step 3: Relate this volume to the tank.
Tank volume = Base area × height rise
Base area of tank = \( \pi R^2 = \pi (0.40)^2 = 0.16\pi \, m^2 \)
Step 4: Find rise in water level.
Rise = \( \dfrac{\text{Volume of inflow}}{\text{Base area}} = \dfrac{0.144\pi}{0.16\pi} = 0.90 \, m \)
Step 5: Convert to cm.
\(0.90 \, m = 90 \, cm\)
Final Answer: Rise in water level = 90 cm.