The rate of change of the height is [tex]\frac{1}{9}[/tex] in/min.
A right circular cylinder is a cylinder that has a closed circular surface having two parallel bases on both the ends and whose elements are perpendicular to its base. It is also called a right cylinder.
Volume of the cylinder = [tex]\pi r^{2}h[/tex]
Rate of change of height is [tex]\frac{dh}{dt}[/tex] = ?
At any time t,
Volume, V = [tex]\pi r^{2}h[/tex]
Since r does not change with time, then r is a constant, so,
V = [tex]\pi (6)^{2}h[/tex]
V = [tex]36\pi h[/tex]
Differentiate both sides with respect to time t,
[tex]\frac{dV}{dt} = 36\pi \frac{dh}{dt}[/tex] -------------(i)
Since [tex]\frac{dV}{dt}[/tex]is given as 4[tex]\pi[/tex] cu.in. per min,
[tex]4\pi = 36\pi \frac{dh}{dt}[/tex]
[tex]\frac{dh}{dt} = \frac{4\pi }{36\pi }[/tex]
[tex]\frac{dh}{dt} = \frac{1 }{9 }[/tex] in/min.
Hence, The rate of change of the height is [tex]\frac{1}{9}[/tex] in/min.
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