1. **Problem:** Water is flowing into a vertical cylindrical tank at the rate of 24 cu.ft/min. The radius of the tank is 24 ft. How fast is the surface rising? 2. **Formula:** The volume of a cylinder is given by $$V = \pi r^2 h$$ where $r$ is the radius and $h$ is the height (surface level) of the water. 3. Since the radius $r=24$ ft is constant, differentiate both sides with respect to time $t$: $$\frac{dV}{dt} = \pi r^2 \frac{dh}{dt}$$ 4. Substitute the known values: $$24 = \pi (24)^2 \frac{dh}{dt}$$ 5. Simplify: $$24 = \pi \times 576 \times \frac{dh}{dt}$$ 6. Solve for $\frac{dh}{dt}$: $$\frac{dh}{dt} = \frac{24}{\pi \times 576}$$ 7. Simplify the fraction: $$\frac{dh}{dt} = \frac{24}{576\pi} = \frac{\cancel{24}}{\cancel{576}\pi} = \frac{1}{24\pi}$$ 8. **Answer:** The surface is rising at a rate of $$\frac{1}{24\pi}$$ ft/min, approximately 0.0133 ft/min.