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23-CS-1 Engineering Economics · May 2018

Question 2 of 5: City Tunnel — Present and Annual Worth

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

National Exams — May 2018 — 11-CS-1 Engineering Economics. Open book; non-communicating calculator permitted. Any four of the five questions constitute a complete paper; all questions are of equal value. Fully worked solutions to all five follow; standard compound-interest factors are used and minor rounding is immaterial.

Question 2: City Tunnel — Present and Annual Worth (25 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Timeline ($t=0$ = end 2018): construction $16M at $t=2\text{–}5$ (ends 2020–2023); operation 2024–2058 = $t=6$ to $t=40$; O&M starts $t=6$ ($1M, +$80k/yr) for 35 years; tolls $8M/yr at $t=6$ to $t=30$ (25 years, to end 2048); $i=6\%$.

(a) Cash-Flow Diagram

Cash flows in millions of dollars; t = 0 is the end of 2018. Down = cost, up = receipt. t 0 2 5 30 40 6 2018 2020 2023 2048 2058 −16 construction, four payments at t = 2, 3, 4, 5 O&M: −1.0 at t = 6, rising −0.08/yr to −3.72 at t = 40 +8 toll revenue, t = 6 to t = 30 (25 years)

Figure 1 — Cash-flow diagram for the tunnel, drawn from the end of 2018 ($t=0$) to the end of 2058 ($t=40$). Four construction outflows of $16M fall at $t=2,3,4,5$ (calendar 2020–2023); the operating phase runs $t=6$ to $t=40$, carrying an arithmetic-gradient O&M cost that starts at $1.0M and grows by $80,000 each year to $3.72M at $t=40$; toll receipts of $8M/yr run only from $t=6$ to $t=30$ (25 years, ending 2048).

(b) Present Worth (t = 0, i = 6%)

Construction: $16M at $t=2\text{–}5$: $16(P/A,6\%,4)(P/F,6\%,1) = 16(3.46511)(0.943396) = \$52.3035$M.

O&M (value at $t=5$): $1(P/A,6\%,35)+0.08(P/G,6\%,35) = 1(14.49825)+0.08(165.74273) = 14.49825+13.25942 = 27.75767$M; to $t=0$: $27.75767(0.747258)=\$20.7421$M.

Tolls (value at $t=5$): $8(P/A,6\%,25) = 8(12.78336) = 102.26688$M; to $t=0$: $102.26688(0.747258)=\$76.4197$M. Combining:

$$PW = -52.3035 - 20.7421 + 76.4197 = \boxed{+\$3.3741\text{M}}$$

(c) Annual Worth 2018–2058 (40 years)

$$AW = PW\,(A/P,6\%,40) = 3.3741(0.066462) = \boxed{+\$0.2242\text{M/yr}\ (\$224{,}200)}$$

(d) Good Investment?

Both PW (+$3.37M) and AW (+$224,200/yr) are positive, so yes—it is a (marginally) good investment: the toll revenue more than covers construction and operating costs at 6%. The margin is thin, so the decision is sensitive to the toll and cost estimates.