Question

The oil company located near Germany wants to install a new pipeline to connect a storage tank to an existing pump 1,000 feet away. Three possible diameters are being considered to make the pipeline, the related costs of which are presented below:

The company is presumed to apply a 10% interest rate on its projects. Both maintenance and the annual cost of pumping are considered as expenses that occur at the end of the year (conventional flows).

For what number of pumping hours per year will the 4 and 6 inch pipelines be equally economical?

You must show in detail how you got your result. It is important that you submit the relevant evidence (cash flow equations, Excel, photo, etc.) in the link provided by this question for correction.

Pipeline diameter	4 pulg.	6 pulg.	8 pulg.
Initial cost	$2,000	$3,000	$4,000
Maintenance cost / year	600	500	400
Cost of pumping per hour	2.70	1.30	1.00
Residual value	100	150	300
Useful life (years)	10	10	10

Answer 1

Let us first see the PV Factors at 10%

Year	PV Factor
0	1
1	0.909
2	0.826
3	0.751
4	0.683
5	0.621
6	0.564
7	0.513
8	0.467
9	0.424
10	0.386
	6.144

For 4 pulg.

Let number of hours be H

Present value = 2,000 + 600 X 6.144 + 2.70H X 6.144 - 100 X 0.386 = 16.5888 H + 5647.8

For 6 pulg

Let number of Hours be H

Present value = 3,000 + 500 X 6.144 + 1.30H X 6.144 - 150 X 0.386 = 7.9872 H + 6014.1

Since hours would be same

16.5888 H + 5647.8 = 7.9872 H + 6014.1
8.6061 H = 366.3
H = 42.5851 Hours

Here is the proof PV of both options is equal

Year	PV Factor	Amount	Present value		Amount	Present value
0	1	2000	2000		3000	3000
1	0.909	714.9797712	649.916612		555.3606306	504.8228132
2	0.826	714.9797712	590.573291		555.3606306	458.7278809
3	0.751	714.9797712	536.9498082		555.3606306	417.0758336
4	0.683	714.9797712	488.3311837		555.3606306	379.3113107
5	0.621	714.9797712	444.0024379		555.3606306	344.8789516
6	0.564	714.9797712	403.248591		555.3606306	313.2233956
7	0.513	714.9797712	366.7846226		555.3606306	284.9000035
8	0.467	714.9797712	333.8955532		555.3606306	259.3534145
9	0.424	714.9797712	303.151423		555.3606306	235.4729074
10	0.386	614.9797712	237.3821917		405.3606306	156.4692034
	6.144		6354.235714			6354.235714