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PROBLEM SET CALCULUS

Allan Abungan

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PROBLEM #1

A horizontal cylindrical tank of length 9 ft. and

radius 5 ft. is filled with oil. At t=0 a plug at

the lowest point of the tank is removed and a

flow results. Find y, the depth of the oil in thetank at any time t while the tank is draining.

The constriction coefficient is k=1/15

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ILLUSTRATION

9

5

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Now in general when the liquid of volume V

escapes through the opening in the container

the rate at which it flows through the opening

is

Thus

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K is the constant that depends on the nature

of the orifice. It is less than one because of

the physically observed fact that the stream

of the escaping liquid contracts slightly as itleaves the container.

To use the relation

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We note that dV is equal to the surface of the

liquid multiplied by dy. Thus

Or

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We have obtained a first degree differential

equation which is separated. To integrate the

left hand side,

Thus the general solution is

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To eliminate c, use the condition that at t=0,

y= 10; then c = 0. thus

Solving for y

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PROBLEM #2

A tank contains 50 gallons of a solution

composed of 90% water and 10% alcohol.A

second solution containing 50% water and

50% alcohol is added to the tank at the rateof 4 gallons per minute. As the second

solution is being added, the tank is being

drained at the rate of 5 gallons per minute.Assuming the solution in the tank is stirred

constantly, how much alcohol is in the tank

after 10 minutes?

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ILLUSTRATION

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Solution Let y be the number of gallons o f

alcohol in the tank at any t ime t. You know

that y=5 when t=0.Because the number of

gallons of solution in the tank at any

time is 50-t and the tank loses 5 gallons of

solution per minute, it must lose

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gallons of alcohol per minute. Furthermore,

because the tank is gaining 2 gallons of

alcohol per minute, the rate of change of

alcohol in the tank is given by

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To solve this linear equation, let

And obtain

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Because t

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Thus the general solution

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Because y=5 when t = 0, you have

Which means that the particular solution is

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