After a factory has been built near a stream, it is learned that the stream occasionally overflows its banks. A hydrologic study indicates that the probability of flooding is about 1 in 8 in any one year. A flood would cause about \(\$ 20,000\) in damage to the factory. A levee can be constructed to prevent flood damage. Its cost will be \(\$ 54,000\) and its useful life is 30 years. Money can be borrowed at 8 percent interest. If the annual equivalent cost of the levee is less than the annual expectation of flood damage, the levee should be built. The annual expectation of flood damage is (1/8) \(\times\) \(20,000=\$ 2,500\). Compute the annual equivalent cost of the levee. a) \(\$ 1,261\) c) \(\$ 4,320\) b) \(\$ 1,800\) d) \(\$ 4,800\)

Step by step solution

01

Understanding given values

We are given the following information: - Probability of flooding in any one year: \(1/8\) - Flood damage cost: \(\$20,000\) - Levee cost: \(\$54,000\) - Useful life of levee: 30 yrs - Interest rate: 8 % We're required to calculate the annual equivalent cost of the levee and compare it with the annual expectation of flood damage (\(\$ 2,500\)) to decide whether the levee should be built.

02

Calculate Annual Equivalent Cost (AEC)

We can calculate the AEC using the following formula: AEC = P × (i × (1 + i)^n) / ((1 + i)^n - 1) Where: - P = Initial cost of the levee - i = Interest rate - n = Useful life of the levee In our case, P = \(\$54,000\), i = 0.08 (8%), and n = 30 years.

03

Substitute and calculate

Now, substituting the values, we get: AEC = \(54,000 \times \dfrac{0.08 \times (1 + 0.08)^{30}}{((1 + 0.08)^{30} - 1)}\) Calculate the AEC step by step: \( (1 + 0.08)^{30} = 10.063 \) AEC = \(54,000 \times \dfrac{0.08 \times 10.063}{10.063 - 1}\) AEC = \(54,000 \times \dfrac{0.8}{9.063}\) AEC ≈ \(4,800\) The AEC is approximately \(\$4,800\). Comparing it with annual expectation of flood damage (\(\$2,500\)), the AEC of the levee is greater. Thus, the levee should not be built. The correct answer is: d) \(\$ 4,800\)

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