Determine the average value for the data in Fig. P. Integration provides a means to compute how much

concentration to give the total mass entering or leaving from t 1 to t 2 . Use numerical integration to evaluate this equation for the data listed below: t , min 10 20 30 35 40 45 50 Q , m 3 min 4 4.8 5.2 5.0 4.6 4.3 4.3 5.0 c , mgm 3 10 35 55 52 40 37 32 34 19.17 The cross-sectional area of a channel can be com- puted as A c = B H y d y where B = the total channel width m, H = the depth m, and y = distance from the bank m. In a similar fashion, the average flow Q m 3 s can be computed as Q = B U yH y d y where U = water velocity ms. Use these relationships and a numerical method to determine A c and Q for the following data: y , m 2 4 5 6 9 H , m 0.5 1.3 1.25 1.8 1 0.25 U , ms 0.03 0.06 0.05 0.13 0.11 0.02

19.18 The average concentration of a substance

¯cgm 3 in a lake where the area A s m 2 varies with depth zm can be computed by integration: ¯c = Z cz A s z d z Z A s z d z where Z = the total depth m. Determine the average con- centration based on the following data: z , m 4 8 12 16 A , 10 6 m 2 9.8175 5.1051 1.9635 0.3927 0.0000 c , gm 3 10.2 8.5 7.4 5.2 4.1

19.19 As was done in Section 19.9, determine the work per-

formed if a constant force of 1 N applied at an angle θ results in the following displacements. Use the cumtrapz function to determine the cumulative work and plot the result versus θ. x , m 1 2.8 3.9 3.8 3.2 1.3 θ , deg 30 60 90 120 150 180

19.20 Compute work as described in Sec. 19.9, but use the

following equations for Fx and θ x: Fx = 1.6x − 0.045x 2 θ x = −0.00055x 3 + 0.0123x 2 + 0.13x The force is in Newtons and the angle is in radians. Perform the integration from x = 0 to 30 m. PROBLEMS 495 FIGURE P19.15 – 8 – 3 – 1 4 7 10 – 8 4 2 4 8 12 – 4 – 2 – 6 1 4 x y 19.21 As specified in the following table, a manufactured spherical particle has a density that varies as a function of the distance from its center r = 0: r , mm 0.12 0.24 0.36 0.49 ρ gcm 3 6 5.81 5.14 4.29 3.39 r , mm 0.62 0.79 0.86 0.93 1 ρ gcm 3 2.7 2.19 2.1 2.04 2 Use numerical integration to estimate the particle’s mass in g and average density in gcm 3 . 19.22 As specified in the following table, the earth’s density varies as a function of the distance from its center r ⫽ 0: r , km 1100 1500 2450 3400 3630 ρ gcm 3 13 12.4 12 11.2 9.7 5.7 r , km 4500 5380 6060 6280 6380 ρ gcm 3 5.2 4.7 3.6 3.4 3 Use numerical integration to estimate the earth’s mass in met- ric tonnes and average density in gcm 3 . Develop vertically stacked subplots of top density versus radius, and bottom mass versus radius. Assume that the earth is a perfect sphere.