this question has several parts that must be completed sequentially. if you skip a part of the question, you…

this question has several parts that must be completed sequentially. if you skip a part of the question, you will not receive any points for the skipped part, and you will not be able to come back to the skipped part.\ntutorial exercise\na cylindrical tank with radius 6 m is being filled with water at a rate of 4 m³/min. how fast is the height of the water increasing (in m/min)?\nstep 1\nlet r represent the radius of the cylindrical tank in m, let h represent the height of the water in the tank in m, and let v represent the volume of the water in the tank in m³. writing an equation for v in terms of r and h gives the following result.\nv = \nwe are given that the radius of the tank is 6 m, and therefore the radius of the column of water that is being measured remains at a constant 6 m. substituting the value r = 6 into the volume equation gives a simplified equation for v in terms of h, as follows.\nv =

this question has several parts that must be completed sequentially. if you skip a part of the question, you will not receive any points for the skipped part, and you will not be able to come back to the skipped part.\ntutorial exercise\na cylindrical tank with radius 6 m is being filled with water at a rate of 4 m³/min. how fast is the height of the water increasing (in m/min)?\nstep 1\nlet r represent the radius of the cylindrical tank in m, let h represent the height of the water in the tank in m, and let v represent the volume of the water in the tank in m³. writing an equation for v in terms of r and h gives the following result.\nv = \nwe are given that the radius of the tank is 6 m, and therefore the radius of the column of water that is being measured remains at a constant 6 m. substituting the value r = 6 into the volume equation gives a simplified equation for v in terms of h, as follows.\nv =

Answer

Explanation:

Step1: Recall volume formula for cylinder

The volume formula for a cylinder is $V=\pi r^{2}h$.

Step2: Substitute given radius value

Given $r = 6$, substituting into $V=\pi r^{2}h$ gives $V=\pi\times6^{2}h= 36\pi h$.

Step3: Differentiate with respect to time

Differentiate both sides of $V = 36\pi h$ with respect to time $t$. Using the chain - rule, $\frac{dV}{dt}=36\pi\frac{dh}{dt}$.

Step4: Solve for $\frac{dh}{dt}$

We know that $\frac{dV}{dt}=4\ m^{3}/min$. So, $4 = 36\pi\frac{dh}{dt}$. Then $\frac{dh}{dt}=\frac{4}{36\pi}=\frac{1}{9\pi}\ m/min$.

Answer:

$\frac{1}{9\pi}\ m/min$