The specific growth rate of a yeast that produces an

is a function of the conversion x A . Find the conversion that will result in the lowest cost system. C is a proportionality constant. Cost = C 1 1 − x A 2 0.6 + 6 1 x A 0.6

7.28 A finite-element model of a cantilever beam subject to

loading and moments Fig. P7.28 is given by optimizing f x, y = 5x 2 − 5x y + 2.5y 2 − x − 1.5y where x = end displacement and y = end moment. Find the values of x and y that minimize f x, y.

7.29 The Streeter-Phelps model can be used to compute the

dissolved oxygen concentration in a river below a point dis- charge of sewage Fig. P7.29, o = o s − k d L o k d + k s − k a e −k a t − e −k d +k s t − S b k a 1 − e −k a t P7.29 where o = dissolved oxygen concentration mgL, o s = oxygen saturation concentration mgL, t = travel time d, L o = biochemical oxygen demand BOD concentration at the mixing point mgL, k d = rate of decomposition of BOD d −1 , k s = rate of settling of BOD d −1 , k a = reaera- tion rate d −1 , and S b = sediment oxygen demand mgLd. As indicated in Fig. P7.29, Eq. P7.29 produces an oxygen “sag” that reaches a critical minimum level o c , some travel time t c below the point discharge. This point is called “critical” because it represents the location where biota that depend on oxygen like fish would be the most stressed. Determine the critical travel time and concentration, given the following values: o s = 10 mgL k d = 0.1 d −1 k a = 0.6 d −1 k s = 0.05 d −1 L o = 50 mgL S b = 1 mgLd 7.30 The two-dimensional distribution of pollutant concen- tration in a channel can be described by cx, y = 7.9 + 0.13x + 0.21y − 0.05x 2 −0.016y 2 − 0.007x y Determine the exact location of the peak concentration given the function and the knowledge that the peak lies within the bounds −10 ≤ x ≤ 10 and 0 ≤ y ≤ 20. 7.31 A total charge Q is uniformly distributed around a ring- shaped conductor with radius a. A charge q is located at a distance x from the center of the ring Fig. P7.31. The force exerted on the charge by the ring is given by F = 1 4π e q Qx x 2 + a 2 32 where e = 8.85 × 10 − 12 C 2 N m 2 , q = Q = 2 × 10 − 5 C, and a = 0.9 m. Determine the distance x where the force is a maximum.

7.32 The torque transmitted to an induction motor is a func-

tion of the slip between the rotation of the stator field and the rotor speed s, where slip is defined as s = n − n R n x a Q q FIGURE P7.31 15 20 8 12 t d 5 4 10 o mgL o o s t c o c FIGURE P7.29 A dissolved oxygen “sag” below a point discharge of sewage into a river. x y FIGURE P7.28 A cantilever beam.