   Chapter 2.1, Problem 52E

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# If a cylindrical tank holds 100, 000 gallons of water, which can be drained from the bottom of the tank in an hour, then Torricelli’s Law gives the volume V of water remaining in the tank after t minutes as V ( t ) = 100.000 ( 1 − 1 60 t ) 2            0 ≤ t ≤ 60 Find the rate at which the water is flowing out of the tank (the instantaneous rate of change of V with respect to t) as a function of t. What are its units? For times t = 0, 10, 20, 30, 40, 50, and 60 min, find the flow rate and the amount of wate remaining in the tank. Summarize your findings in a sentence or two. At what time is the flow rate the greatest? The least?

To determine

To find:

(a)The instantaneous rate of change of V with respect to t that is flow rate& its units.

(b)The time at which the flow rate is greatest and least.

Explanation

To find the rate at which the water is flowing out of the tank need to find instantaneous rate of change with respect to t.

To find instantaneous rate of change with respect to t differentiate V with respect to t.

Given:

Vt= 1000001-160t2  0 t60

Calculation:

Part (a):By using chain rule of differentiation, we have

V't=100000*21-160tddt (1-160t)

V't=100000*21-160t (-160)

V't=- 1000031-160t

V't=-3333.33+55.56 t  gallons/min

Therefore, flow rate =V't=-3333.33+55.56 t  gallons/min

Part (b):

To find the flow rate V’ (t) and the amount of water remaining in the tank V (t) substitute the given t values.

Substitute t = 0 in V’ (t) and V (t).

Flow rate:  V'0= -3333.33+55.560=-3333.33 gallons /min

Remaining water: V0=100000(1-160*0)2=100000 gallons left

Substitute t = 10 in V’ (t) and V (t).

Flow rate: V'10= -3333.33+55.5610= -2777.73 gallons/min

Remaining water: V10=1000002536= 69444.44 gallons  left

Substitute t = 20 in V’ (t) and V (t)

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