Answer:
The rate of change of height = 19.34 ft/min
Step-by-step explanation:
The Volume of a cylinder (v) = πr²h ....................... equation 1
Where r = radius of the base, h = height of the cylinder.
Using chain rule,
dv/dt = (dv/dr) . (dr/dt) .................... equation 2
Where dv/dt = rate of change of the volume, dv/dr = differentiation of the volume of cylinder with respect to the radius, dr/dt = rate of change of the radius.
dv/dt = 841 ft³/min,
dr/dt = 4 ft/min,
dv/dr = differentiation of equation 1 with respect to r = 2πrh
where h = 9 ft. ∴ dv/dr = 2(3.143×9)r
dv/dr = 56.574r
Substituting these values into equation 2,
841 = 56.57r(4)
841 = 226.296r
226.296r = 841
dividing both sides of the equation by the coefficient of r,
226.296r/226.296 = 841/226.296
r = 3.72 ft.
Also using chain rule for the rate of change of the height,
dv/dt = (dh/dt)(dv/dh)........................ equation 3
∴ dh/dt = (dv/dt)/(dv/dh)...................... equation 4
Where dv/dt =rate of change of volume= 841 ft³/min,
dv/dh = differentiation of volume with respect to height = πr² (differentiate equation 1 with respect to height.)
dv/dh = 3.143(3.72)² = 43.49 ft
applying the values above into equation 4
∴dh/dt = 841/43.49 = 19.34 ft/min
The rate of change of height = 19.34 ft/min
