8-2 ELECTRICAL СОMPU ТER FE AND PRACTICE PROBLEMS 11. A tank with a cross-sectional area of 12 m2 and a height of 10 m is filled with oil at a rate of 2.2 m3/min The density of the oil is 847 kg/m3. The oil leaks out of the tank from an open tap at the bottom of the tank The leak rate is 0.11t m3/min, where t is the time in elapsed minutes. Most nearly, what is the level of oil in the tank after the tank has been filled for 6 hr? 7. What is the solution to the following differential equation if x= - 1 at t- 0 , and dx/ dt= 0 at t 0? dax 1 8x 5 4- dt d2 4t e (A) -4t +4te (А) 5.2 m 5 -2cos 2t sin 2t) + 3 (B) (В) 6.6 m (C) 7.1 m -4t e 4t C) 4te 8 (D) 9.4 m 4t te -4t е 8 (D) x 8. In the following differential equation with the initial condition (0) = 12, what is the value of (2)? dx 4x 0 dt (A) 3.4 x 10-3 (В) 4.0 x 10-3 (С) 5.1 х 10-3 (D) 6.2 x 10-3 9. What are the three general Fourier coefficients for the sawtooth wave shown? f(t) 2 1 t -1 (А) аg — 0, а,р, 3D 0, b, Tn 1 -1 (B) ao аn — 0, bл 2 тп 1 (C) ao 1, an 1, b тп 1 1 (D) ao Иn 2 п 2 TTn 10. The values of an unknown function follow a Fibo- nacci number sequence. It is known that f(1) = 4 and f(2) 1.3. What is f(4)? (А) - 4.1 (В) 0.33 (C) 2.7 (D) 6.6 P PI ppi 2p ass.com 8-5 DIFFERENTIAL E QUATIONS Use the second-order difference equation 8. This is a first-order, linear, homogeneous differential equation with characteristic equation r+ 4 0. f(k) f(k-1) f(k 2) f(3) f(2)f(1) 1.34 4x 0 4t = 5.3 f(4) f(3)(2) = 5.31.3 x(0)= ne-4)(0) = 6.6 12 12 The answer is (D). 12e 4t x(2) 12e-4)(2) 11. The general equation for the unsteady-state mass balance is 12e-8 = 4.03 x 10 (4.0 x 103) min mout 'accumulation The answer is (B). The mass flow rates can be converted to volumetric flow rates 9. By inspection, f(t) t, with the period T 1. The angular frequency is Tin T out 'accumulation pQaccumulation in 2т 2т Qin Qaccumulation Qout 2T 1 Т The volume of oil accumulating in the tank changes with time The is average dV T Qinout (1/T)(t) dt (1/T)t 0 2 ао— dt 3 3 2.2 min 0.11t min 2 Since the cross-sectional area is constant, the volume of oil accumulating can be expressed in terms of the leak rate The general a term is (2/T)f(t)cos(nt) dt ат 3 m dh A. - 0.011t = 2.2 t cos (27nt) dt 2 dt min min 3 2.2 min 0.011t 0 = dh min A A dt The general b term is 3 m 0.011t 2.2 min Т b(2/T)f(t)sin(nwot)di min 12 m2 0.1833 m/min - 0.0009167t m/min 12 m2 t sin(2Tnt) dt = 2 1 The answer is (B). 10. The value of a number in a Fibonacci sequence is the sum of the previous two numbers in the sequence РPI Oрpі 2ра 8-6 FE ELECTRICAL AND COMPUTER PRACT1CE PROBLEMS Integrate both sides with respect to time. dh m |dt min 0.1833 min dt 0.0009167t dt 16h 0.0009167 t2 min m. 0.1833 h 2 min min = (0.1833 m)(6 h)(60- 0.0009167 min 60 h 2 = 6.585 m (6.6 m) The answer is (B). ppi2p as PPI Mathematics Mathematics Mathematics