Question

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tutorial exercise
a 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)?
step 1
let 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.
v =
we 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.
v =

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:

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