I need the solution with more details and fourmlas The city of Peachtree is comp

Question

I need the solution with more details and fourmlas The city of Peachtree is comparing the following two plans for supplying water to a newly developed subdivision: Plan A will manage requirements for the next 15 years; at the end of that period, the initial cost of $1,500,000 will have to be doubled to meet the requirements of subsequent years. The facilities installed in years 0 and 15 may be considered permanent; however, certain supporting equipment will have to be replaced every 30 years from the installation dates at a cost of $200,000. Operating costs are $91,000 a year for the first 15 years and $182,000 thereafter. Beginning in the 21st year, they will increase by $3,000 a year. Plan B will supply all requirements for water indefinitely into the future, although it will operate at only half capacity for the first 15 years. Annual costs over this period will be $105,000 and will increase to $155,000 beginning in the 16th year. The initial cost of Plan B is $1,950,000; the facilities can be considered permanent, although it will be necessary to replace $350,000 of equipment every 30 years after the initial installation. The city will charge the subdivision the use of water calculated on the equivalent annual cost. At an interest rate of 10%, determine the equivalent annual cost for each plan and make a recommendation to the city as to the amount that should be charged to the subdivision.

Explanation / Answer

The city should select the plan that gives lower average annual cost per unit of water supplied.

Average annual cost per unit =The cumulative cost incurred till the year considered / total units of water supplied

Cumulative cost incurred till any year = Sum of total expenses incurred in all the years till that year

Total expenses incurred = Capital expenditure + Operational expense + Interest incurred

Capital expenditure includes cost if initial installation of equipment and cost of replacement of equipment

Let's assume that the system will supply 1 million units of water at its full capacity. Then as per plan B, for the first 15 years, only 500,000 units of water will be supplied per annum. Let's also assume that interest will not be during the operatinal life of the supply system. Therefore,

Interest incurred = 10% of Cumulative total expense till the previous year

Based on the formulae defined above, let's compare the average annual cost per unit for the two plans:

     2,982,437.09

It is visible from the table that 16th year onwards, plan B costs lower than plan A. Therefore, the city should adopt plan B.  

Years Plan A Plan B 1                       1.74                       4.50 2                       1.00                       2.58 3                       0.77                       1.96 4                       0.65                       1.67 5                       0.59                       1.51 6                       0.56                       1.42 7                       0.54                       1.37 8                       0.53                       1.35 9                       0.53                       1.34 10                       0.53                       1.35 11                       0.54                       1.37 12                       0.55                       1.39 13                       0.57                       1.43 14                       0.59                       1.48 15                       0.81                       1.53 16                       0.84                       0.80 17                       0.88                       0.84 18                       0.92                       0.88 19                       0.96                       0.92 20                       1.01                       0.97 21                       1.06                       1.03 22                       1.12                       1.08 23                       1.18                       1.15 24                       1.25                       1.22 25                       1.32                       1.29 26                       1.40                       1.37 27                       1.49                       1.46 28                       1.58                       1.55 29                       1.69                       1.65 30                       1.80                       1.77 31                       1.92                       1.89 32                       2.05                       2.02 33                       2.19                       2.16 34                       2.34                       2.31 35                       2.50                       2.48 36                       2.68                       2.65 37                       2.87                       2.84 38                       3.08                       3.05 39                       3.31                       3.27 40                       3.55                       3.51 41                       3.81                       3.77 42                       4.10                       4.06 43                       4.41                       4.36 44                       4.74                       4.69 45                       5.10                       5.05 46                       5.49                       5.44 47                       5.92                       5.86 48                       6.38                       6.31 49                       6.88                       6.81 50                       7.42                       7.34 51                       8.00                       7.92 52                       8.64                       8.55 53                       9.32                       9.23 54                     10.07                       9.96 55                     10.88                     10.76 56                     11.76                     11.63 57                     12.71                     12.57 58                     13.74                     13.60 59                     14.87                     14.70 60                     16.08                     15.91 61                     17.40                     17.22 62                     18.84                     18.64 63                     20.40                     20.18 64                     22.09                     21.85 65                     23.93                     23.67 66                     25.93                     25.65 67                     28.10                     27.79 68                     30.46                     30.12 69                     33.02                     32.66 70                     35.81                     35.41 71                     38.84                     38.41 72                     42.13                     41.66 73                     45.71                     45.21 74                     49.61                     49.06 75                     53.84                     53.24 76                     58.45                     57.80 77                     63.46                     62.76 78                     68.92                     68.15 79                     74.86                     74.02 80                     81.31                     80.40 81                     88.35                     87.35 82                     96.00                     94.92 83                  104.33                  103.16 84                  113.40                  112.12 85                  123.28                  121.88 86                  134.03                  132.52 87                  145.74                  144.09 88                  158.50                  156.70 89                  172.39                  170.44 90                  187.53                  185.41 91                  204.02                  201.71 92                  221.98                  219.47 93                  241.56                  238.82 94                  262.89                  259.91 95                  286.14                  282.89 96                  311.48                  307.94 97                  339.09                  335.24 98                  369.20                  365.01 99                  402.02                  397.45 100                  437.81                  432.83 101                  476.82                  471.40 102                  519.36                  513.46 103                  565.76                  559.32 104                  616.35                  609.34 105                  671.53                  663.89 106                  731.72                  723.39 107                  797.37                  788.30 108                  868.99                  859.10 109                  947.13                  936.34 110               1,032.37               1,020.61 111               1,125.38               1,112.56 112               1,226.87               1,212.89 113               1,337.62               1,322.37 114               1,458.48               1,441.85 115               1,590.38               1,572.25 116               1,734.34               1,714.56 117               1,891.47               1,869.90 118               2,062.98               2,039.46 119               2,250.22               2,224.56 120               2,454.61               2,426.63 121               2,677.76               2,647.23 122               2,921.40               2,888.09 123               3,187.42               3,151.07 124               3,477.89               3,438.22 125               3,795.07               3,751.79 126               4,141.45               4,094.22 127               4,519.73               4,468.18 128               4,932.86               4,876.60 129               5,384.09               5,322.67 130               5,876.94               5,809.91 131               6,415.29               6,342.11 132               7,003.37               6,923.47 133               7,645.78               7,558.56 134               8,347.60               8,252.37 135               9,114.35               9,010.37 136               9,952.07               9,838.53 137            10,867.37            10,743.38 138            11,867.48            11,732.09 139            12,960.32            12,812.45 140            14,154.52            13,993.03 141            15,459.55            15,283.17 142            16,885.76            16,693.10 143            18,444.44            18,234.00 144            20,148.00            19,918.11 145            22,009.95            21,758.82 146            24,045.12            23,770.77 147            26,269.71            25,969.97 148            28,701.44            28,373.95 149            31,359.69            31,001.87 150            34,265.69            33,874.71 151            37,442.65            37,015.42 152            40,915.95            40,449.09 153            44,713.38            44,203.18 154            48,865.34            48,307.77 155            53,405.09            52,795.71 156            58,369.03            57,703.01 157            63,796.98            63,069.02 158            69,732.53            68,936.84 159            76,223.36            75,353.60 160            83,321.66            82,370.91 161            91,084.55            90,045.22 162            99,574.54            98,438.32 163          108,860.02          107,617.85 164          119,015.86          117,657.80 165          130,124.01          128,639.20 166          142,274.15          140,650.69 167          155,564.43          153,789.32 168          170,102.30          168,161.30 169          186,005.36          183,882.89 170          203,402.34          201,081.35 171          222,434.14          219,895.98 172          243,255.01          240,479.27 173          266,033.81          262,998.14 174          290,955.37          287,635.32 175          318,222.05          314,590.86 176          348,055.37          344,083.75 177          380,697.85          376,353.75 178          416,415.01          411,663.35 179          455,497.54          450,299.91 180          498,263.70          492,578.07 181          545,061.95          538,842.31 182          596,273.82          589,469.80 183          652,317.05          644,873.52 184          713,649.03          705,505.65 185          780,770.62          771,861.32 186          854,230.23          844,482.68 187          934,628.37          923,963.41 188      1,022,622.64      1,010,953.58 189      1,118,933.13      1,106,165.08 190      1,224,348.41      1,210,377.48 191      1,339,732.04      1,324,444.47 192      1,466,029.70      1,449,300.95 193      1,604,277.06      1,585,970.78 194      1,755,608.35      1,735,575.24 195      1,921,265.76      1,899,342.34 196      2,102,609.72      2,078,617.00 197      2,301,130.24      2,274,872.21 198      2,518,459.21      2,489,721.26 199      2,756,384.00      2,724,931.10 200      3,016,862.29

     2,982,437.09

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