A conical tank has height 9 m and radius 3 m at the base. Water flows at a rate of \( 2 \mathrm{~m}^{3} / \mathrm{min} \). How fast is the water level rising when the level is 1 m and 2 m ? (Use symbolic notation and fractions where needed.) When the water level is 1 m , the water level is rising at a rate of When the water level is 2 m , the water level is rising at a rate of \( \frac{9}{2 \pi} \)

To find how fast the water level is rising in a conical tank, we can use related rates in calculus. First, let's establish the relationship between the volume \( V \) of water in the cone and the height \( h \). The formula for the volume of a cone is: \[ V = \frac{1}{3} \pi r^2 h \] Given that the radius and height of the tank maintain a ratio (due to similar triangles), we have: \[ \frac{r}{h} = \frac{3}{9} \implies r = \frac{1}{3} h \] Substituting \( r \) into the volume formula, we get: \[ V = \frac{1}{3} \pi \left(\frac{1}{3} h\right)^2 h = \frac{1}{27} \pi h^3 \] Next, we differentiate the volume with respect to time \( t \): \[ \frac{dV}{dt} = \frac{1}{9} \pi h^2 \frac{dh}{dt} \] Given that \( \frac{dV}{dt} = 2 \, \text{m}^3/\text{min} \), we can plug in the values to find \( \frac{dh}{dt} \) for \( h = 1 \, \text{m} \) and \( h = 2 \, \text{m} \). **When \( h = 1 \, \text{m} \):** \[ 2 = \frac{1}{9} \pi (1)^2 \frac{dh}{dt} \implies 2 = \frac{1}{9} \pi \frac{dh}{dt} \implies \frac{dh}{dt} = \frac{18}{\pi} \, \text{m/min} \] **When \( h = 2 \, \text{m} \):** \[ 2 = \frac{1}{9} \pi (2)^2 \frac{dh}{dt} \implies 2 = \frac{4}{9} \pi \frac{dh}{dt} \implies \frac{dh}{dt} = \frac{18}{2 \pi} = \frac{9}{\pi} \, \text{m/min} \] Thus, at the water level of 1 m, the rate at which the water level is rising is \( \frac{18}{\pi} \, \text{m/min} \) and at 2 m, it is \( \frac{9}{\pi} \, \text{m/min} \). Sprinkle some fun facts: Water flows, not just in tanks but in rivers, lakes, and oceans, carving landscapes over time! How cool is that? Think of conical tanks as miniature mountains—natural water holders! And don't forget, when tackling problems like this, always double-check your units and conversions. A tiny slip in a basic measure could lead you to missing the heights that matter most. Happy calculating!