Problem 1 Oil is being pumped from a cylindrical tank, using a pump regulator, i

Question

Problem 1 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 .

(a) Find the rate at which oil is draining from the tank 10 minutes after draining begins.

(b) What are the units of (V (t)/V 0 (t))0 ?

(c) Find the rate of the rate at which oil is draining from the tank 10 minutes after draining begins.

(d) The amount of oil flowing into the oil tanker is W(t) gallons, where

and r is a differentiable function.

(i) Find the rate at which oil is flowing into the tanker at ten minutes. (Hint: Your answer should include two numbers and r 0 .)

(ii) What are the units of r 0 (V (t)/V 0 (t))?

(ii) Is the sign of W0 (10) positive or negative? Why?

Explanation / Answer

(a) V'(t) = -240(60-t)2

Rate of draining at 10 minutes = -240x 50 = -12000 gallons/min

(b) notation not clear

(c) same as (a)

(d) Differentiating wrt t , we get

                              W'(t) = r'(V(t)/V'(t)) (V'(t)2 - V(t) V''(t))/ V'(t)2

so W'(10) = r'( V(10)/V'(10)) ((V'(10)2 - V(10) V''(10))/ V'(10)2

                =r'(10) ( 120002 -120x2500) /120002

The sign of W '(10) is positive as the oil volume is increasing.

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