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SCALCET8 3.9.005.MI. A cylindrical tank with radius 5 m is being filled with water at a rate of 4 m3/min. How fast is the height of the water increasing

Sagot :

Answer:

The height of the water is increasing at a rate of 0.05m/min.

Step-by-step explanation:

Volume of a cylinder:

The volume of a cylinder, with radius r and height h, is given by:

[tex]V = \pi r^2h[/tex]

Radius 5 m

This means that [tex]r = 5[/tex], and so:

[tex]V = 25\pi h[/tex]

How fast is the height of the water increasing?

We have to differentiate V and h implictly in function of t. So

[tex]\frac{dV}{dt} = 25\pi\frac{dh}{dt}[/tex]

Being filled with water at a rate of 4 m3/min

This means that [tex]\frac{dV}{dt} = 4[/tex]. The questions asks [tex]\frac{dh}{dt}[/tex]. So

[tex]\frac{dV}{dt} = 25\pi\frac{dh}{dt}[/tex]

[tex]4 = 25\pi\frac{dh}{dt}[/tex]

[tex]\frac{dh}{dt} = \frac{4}{25\pi}[/tex]

[tex]\frac{dh}{dt} = 0.05[/tex]

The height of the water is increasing at a rate of 0.05m/min.