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EXAMPLE 9.7 A practical coil is connected in series between two voltage sources v1 5 cos !1 t and v2 10 cos !2 t 608 such that the sources share the same reference node. See Fig. 9-54. The voltage di erence across the terminals of the coil is therefore v1 v2 . The coil is modeled by a 5-mH inductor in series with a 10- resistor. Find the current i t in the coil for (a) !1 !2 2000 rad/s and (b) !1 2000 rad/s, !2 2!1 . p (a) The impedance of the coil is R jL! 10 j10 10 2 458 . The phasor voltage between its terminals is p V V1 V2 5 10 608 j5 3 V. The current is p V j5 3 j8:66 % 0:61 1358 A I p Z 10 2 458 14:14 458 i 0:61 cos 2000t 1358

(x y)

(b) Because the coil has di erent impedances at !1 2000 and !2 4000 rad/s, the current may be represented in the time domain only. By applying superposition, we get i i1 i2 , where i1 and i2 are currents due to v1 and v2 , respectively. V1 5 0:35 458 A; i1 t 0:35 cos 2000t 458 Z1 10 j10 V 10 608 0:45 3:48 A; i2 t 0:45 cos 4000t 3:48 I2 2 Z2 10 j20 i i1 i2 0:35 cos 2000t 458 0:45 cos 4000t 3:48 I1

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The project totaled 100 KLOC. Twenty programmers worked on the project for a year. This year included the whole effort for the requirements, design, implementation, testing, and delivery phases. Assume that there are about 240 workdays in a year (20 days a month for 12 months, no vacations). The productivity is 100,000 LOC / 20 240 days = 20.8 LOC/programmer-day.

2.72. Let X and Y be random variables having joint density function f (x, y) (2p) 1 e (x2 y2), ` x `, ` y ` . If R and are new random variables such that X R cos , Y R sin , show that the density function of R is g(r) e re 0

f (x, y)

From Table 9-1, vL !LI cos !t 908 100 cos 2000t 908 (V). vL 100 sin 2000t 1808 100 sin 2000t As a sine function, V

2.74. Let X and Y be identically distributed independent random variables with density function f (t) Find the density function of X e 1 0 0 t 1 otherwise

A series circuit, with R 10 and L 20 mH, has current i 2:0 sin 500t (A). Obtain total voltage v and the angle by which i lags v.

2.75. Let X and Y be identically distributed independent random variables with density function f (t) Find the density function of X e e 0

2.77. If the independent random variables X1 and X2 are identically distributed with density function f (t) find the density function of X1 X2. e te 0

CHAP. 9]

Vic Basili and Dieter Rombach developed this approach at the University of Maryland. GQM stands for goals, questions, and metrics. The idea is to rst identify the goals of the approach. Next, questions are developed related to these goals. Finally, metrics are developed to measure the attributes related to the questions.

2.78. Two points are to be chosen at random on a line segment whose length is a three line segments thus formed will be the sides of a triangle. 0. Find the probability that the

By the methods of Example 9.1, 500 20 10 3 458 arctan 10 q v I R2 !L 2 sin !t 28:3 sin 500t 458 It is seen that i lags v by 458.

2.79. It is known that a bus will arrive at random at a certain location sometime between 3:00 P.M. and 3:30 P.M. A man decides that he will go at random to this location between these two times and will wait at most 5 minutes for the bus. If he misses it, he will take the subway. What is the probability that he will take the subway 2.80. Two line segments, AB and CD, have lengths 8 and 6 units, respectively. Two points P and Q are to be chosen at random on AB and CD, respectively. Show that the probability that the area of a triangle will have height AP and that the base CQ will be greater than 12 square units is equal to (1 ln 2) > 2.

2.81. Suppose that f (x) c > 3x, x 1, 2, c, is the probability function for a random variable X. (a) Determine c. (b) Find the distribution function. (c) Graph the probability function and the distribution function. (d) Find P(2 X 5). (e) Find P(X 3). 2.82. Suppose that f (x) e cxe 0

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