Oil is being pumped from a cylindrical tank, using a pump regulator, into an oil

Question

Oil is being pumped from a cylindrical tank, using a pump regulator, into an oil tanker for transportation. The tank has V(t) gallons of oil left t minutes after draining begins, where V(t) = 120(60 - t)^2. Find the rate at which oil is draining from the tank 10 minutes after draining begins. What are the units of (V(t)/V'(t))'? Find the rate of the rate at which oil is draining from the tank 10 minutes after draining begins. The amount of oil flowing into the oil tanker is W(t) gallons, where W(t) = r (V(t)/V'(t)). and r is a differentiable function. Find the rate at which oil is flowing into the tanker at ten minutes. What are the units of r'(V(t)/V'(t))? Is the sign of W'(10) positive or negative? Why?

Explanation / Answer

solution-

V(t)=120(60-t)2

differentiating wrt time

V'(t)=-240(60-t)

at t=10,

V'(t)=-240*50=-12000gallon/sec

b) [V(t)/V'(t)]'=((volume/(volume/time))'=d/dt (time) = constant

hence it is unitless

c) V(t)=120(60-t)2

V'(t)=-240(60-t)

V''(t)=240.gallon/sec2

d) i) W'(t)=r' *V(t)/V'(t)+r* d/dt(V(t)/V'(t))

W'(t)=r'(-240*(60-t))+r*.5 ; d/dt(V(t)/V'(t)) =0.5 3

at t=10

W'(10)=-r'*240*50+r*0.5

ii) units of   r' *V(t)/V'(t) is same as of W'(t)=gallon/sec

iii) sign will depend upon function r

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