) Applied computational problem: A community has a cylindrical water tank with a

Question

) Applied computational problem: A community has a cylindrical water tank with a drain at the base. It is filled to a depth of 15 m. When the drain is opened, the water will flow fast when the tank is full and slow down as it continues to drain. When draining, the water level drops at the following rate.

‘c’ is a constant taking into account shape of the hole and cross-sectional area of the tank and = 0.06. Water depth is in meters (y) and time (t) is in minutes. If the drain fails, how long will it take for the tank to drain and the community to be out of water? Use Euler’s Method in a spreadsheet, yi = 15m, ti = 0 and h = 2 minutes. If the step size is halved to 1 minute, how does your answer change? Include your spreadsheet print outs and charts for both scenarios.

dy dt cyy

Explanation / Answer

for h=2 min it will take 120 min to empty the tank

f(x,y)= -c(sqrt(y))

Here is the spread sheet of both h values

for h = 1min

it will take 125 min to empty the tank

x0 = 0 y0 = 15 h= 2 n x(n) y(n) f(x(n),y(n)) y(n+1)=y(n) + h*f(x(n),y(n)) 0 0 15 -0.232379001 14.535242 1 2 14.53524 -0.228750675 14.07774065 2 4 14.07774 -0.225121892 13.62749686 3 6 13.6275 -0.221492638 13.18451159 4 8 13.18451 -0.217862897 12.7487858 5 10 12.74879 -0.214232651 12.32032049 6 12 12.32032 -0.210601885 11.89911672 7 14 11.89912 -0.206970578 11.48517557 8 16 11.48518 -0.203338713 11.07849814 9 18 11.0785 -0.199706268 10.67908561 10 20 10.67909 -0.196073221 10.28693916 11 22 10.28694 -0.192439552 9.902060061 12 24 9.90206 -0.188805234 9.524449594 13 26 9.52445 -0.185170242 9.15410911 14 28 9.154109 -0.18153455 8.79104001 15 30 8.79104 -0.177898128 8.435243754 16 32 8.435244 -0.174260947 8.08672186 17 34 8.086722 -0.170622972 7.745475916 18 36 7.745476 -0.166984171 7.411507574 19 38 7.411508 -0.163344505 7.084818564 20 40 7.084819 -0.159703935 6.765410695 21 42 6.765411 -0.156062419 6.453285857 22 44 6.453286 -0.15241991 6.148446037 23 46 6.148446 -0.148776361 5.850893314 24 48 5.850893 -0.145131719 5.560629875 25 50 5.56063 -0.141485927 5.277658021 26 52 5.277658 -0.137838924 5.001980174 27 54 5.00198 -0.134190643 4.733598888 28 56 4.733599 -0.130541013 4.472516863 29 58 4.472517 -0.126889955 4.218736952 30 60 4.218737 -0.123237385 3.972262183 31 62 3.972262 -0.119583209 3.733095765 32 64 3.733096 -0.115927325 3.501241114 33 66 3.501241 -0.112269622 3.27670187 34 68 3.276702 -0.108609975 3.05948192 35 70 3.059482 -0.104948249 2.849585422 36 72 2.849585 -0.101284291 2.647016841 37 74 2.647017 -0.097617932 2.451780977 38 76 2.451781 -0.093948984 2.26388301 39 78 2.263883 -0.090277233 2.083328544 40 80 2.083329 -0.086602441 1.910123662 41 82 1.910124 -0.082924334 1.744274994 42 84 1.744275 -0.079242602 1.58578979 43 86 1.58579 -0.075556887 1.434676015 44 88 1.434676 -0.071866777 1.290942461 45 90 1.290942 -0.068171789 1.154598882 46 92 1.154599 -0.064471358 1.025656166 47 94 1.025656 -0.060764811 0.904126545 48 96 0.904127 -0.057051341 0.790023862 49 98 0.790024 -0.053329972 0.683363919 50 100 0.683364 -0.049599497 0.584164924 51 102 0.584165 -0.04585841 0.492448105 52 104 0.492448 -0.042104788 0.408238529 53 106 0.408239 -0.038336128 0.331566273 54 108 0.331566 -0.034549075 0.262468123 55 110 0.262468 -0.030738986 0.200990152 56 112 0.20099 -0.026899155 0.147191842 57 114 0.147192 -0.023019353 0.101153135 58 116 0.101153 -0.019082748 0.062987638 59 118 0.062988 -0.015058403 0.032870832 60 120 0.032871 -0.010878189 0.011114454 61 122 0.011114 -0.006325507 -0.001536559

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