Water is flowing into a right circular conical vessel, 45 cm deep and 27 cm in diameter at the rate of 11 cc per minute. How fast is the water level rising when the water is 30 cm deep?

The correct option is 0.043cm/minute.

To find how fast the water level is rising, we can relate the volume of the water in the conical vessel to its depth using the geometric properties of a cone.

Step 1: Identify the given values
Let $H$ be the total depth of the vessel and $D$ be the top diameter:
$H=45\text{cm}$
$D=27\text{cm}$
The radius at the top of the vessel, $R$, is half of the diameter:
$R=\frac{27}{2}=13.5\text{cm}$
The rate at which water is flowing into the vessel is the rate of change of volume:
$\frac{dV}{dt}=11{\text{cm}}^3\text{/minute}$

Step 2: Express the radius in terms of height
At any instant, let the radius of the water surface be $r$ and the depth of the water be $h$. Using the property of similar triangles inside the cone:
$\frac{r}{h}=\frac{R}{H}=\frac{13.5}{45}=\frac{3}{10}$
Solving for $r$ gives:
$r=\frac{3h}{10}$

Step 3: Write the volume equation in terms of height
The volume $V$ of a cone is given by:
$V=\frac{1}{3}\pi r^{2}h$
Substitute the expression for $r$ from Step 2 into the volume formula:
$V=\frac{1}{3}\pi {\left(\frac{3h}{10}\right)}^{2}h=\frac{1}{3}\pi \left(\frac{9h^2}{100}\right)h=\frac{3\pi h^3}{100}$

Step 4: Differentiate both sides with respect to time
Taking the derivative with respect to time $t$:
$\frac{dV}{dt}=\frac{3\pi }{100}\times 3h^2\frac{dh}{dt}$
$\frac{dV}{dt}=\frac{9\pi h^2}{100}\frac{dh}{dt}$

Step 5: Substitute the given values to find $\frac{dh}{dt}$
We want to find the rate of change of depth when the water depth $h=30\text{cm}$:
$11=\frac{9\pi {\left(30\right)}^2}{100}\frac{dh}{dt}$
$11=\frac{9\pi \times 900}{100}\frac{dh}{dt}$
$11=81\pi \frac{dh}{dt}$
Solve for $\frac{dh}{dt}$:
$\frac{dh}{dt}=\frac{11}{81\pi }$

Using the approximation $\pi \approx 3.14159$:
$\frac{dh}{dt}\approx \frac{11}{81\times 3.14159}\approx \frac{11}{254.469}\approx 0.0432\text{cm/minute}$

Rounding to three decimal places, the water level is rising at a rate of 0.043 cm/minute.