1. Let V (t) be the volume of water in a tank, in liters, at time t, in seconds.
ID: 2851142 • Letter: 1
Question
1. Let V (t) be the volume of water in a tank, in liters, at time t, in seconds.
(a) What are the meaning and units of dV/dt ?
(b) The tank is full at time t0, so that V (t0) > 0. At some later time t1 > t0 a drain is opened 20 cm above the bottom of the tank(which is taller than 20 cm) emptying water from the tank. Is dV dt positive, negative or zero at: i. t with t0 < t < t1 ii. After the faucet is turned on but before the water reaches 20 cm above the bottom of the tank iii. After the water reaches 20 cm above the bottom of the tank.
2. Let f(t) be the amount of rain, in cm, that has fallen since midnight, with t in hours. Interpret the following in practical terms, giving units.
(a) f(7) = 2.4
(b) f' (7) = 0.21
(c) f ^?1 (3) = 12.3
(d) (f^ ?1 )'(3) = 86
Explanation / Answer
(a) dv/dt refers to the rate of change of volume of water in the tank. Its unit is liters/sec.
(b) (i) dv/dt is negative when t1>t0
(ii) dv/dt is positive after the faucet is turned on but before the water reaches 20cm above the bottom of the tank
(iii) dv/dt =0 After the water reaches 20 cm above the bottom of the tank.
2. (a) f(7) =2.4 means the amount of rain that has fallen for 7 hours since midnight is 2.4cm
(b) f'(7) =0.21 means the rate of rainfall for 7 hours is 0.21 cm/hour
(c) it means that f(12.3) =3 That is the amount of rain fall after 12.3 hours is 3 cm
(d) The rate of change of rain fall after 86 hours is 3 cm/hour
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