Capitalized cost refers to the present value of a sequence of yearly costs. It involves computing the total present value of the stream of costs using a given interest rate. It is computed for items that last for over one year and require maintenance after a period.
To find the capitalized cost of a permanent roadside historical marker that has a first cost of $78,000 and a maintenance cost of $3,500 once every five years at an interest rate of 8% per year:
Step 1: Determine the total number of years the costs will occur. Since the maintenance cost occurs every five years, and the useful life of the roadside marker is infinite, assume the roadside marker will last 100 years. Therefore, the cost of maintaining it will occur every 5 years for a total of 20 times.
Step 2: Calculate the present value of each maintenance cost. Use the formula PV = FV/ (1 + r)n where FV is the future value, r is the interest rate and n is the number of periods (years). Present value of each maintenance cost = $3,500/(1 + 0.08)5 = $2,160.36
Step 3: Calculate the present value of the first cost. Since it occurs in year 0, the present value is equal to the first cost. PV of first cost = $78,000
Step 4: Calculate the capitalized cost using the formula: Capitalized cost = PV of first cost + (PV of each maintenance cost * number of maintenance costs) Capitalized cost = $78,000 + ($2,160.36 x 20)
Capitalized cost = $123,207.20
The capitalized cost of a permanent roadside historical marker that has a first cost of $78,000 and a maintenance cost of $3,500 once every 5 years at an interest rate of 8% per year is $123,207.20.
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We consider three different hash functions which produce outputs of lengths 64, 128 and 160 bit. After how many random inputs do we have a probability of ε = 0. 5 for a collision? After how many random inputs do we have a probability of ε = 0. 1 for a collision?
For ε = 0.1, approximately 2.147 random inputs are needed for a collision. The number of inputs required for the hash functions producing outputs of lengths 128 and 160 bits using the same formula.
To determine the number of random inputs needed to achieve a specific probability of collision, we can use the birthday paradox principle. The birthday paradox states that in a group of people, the probability of two individuals having the same birthday is higher than expected due to the large number of possible pairs.
The formula to calculate the approximate number of inputs required for a given probability of collision (ε) is:
n ≈ √(2 * log(1/(1 - ε)))
Let's calculate the number of inputs needed for ε = 0.5 and ε = 0.1 for each hash function:
For a hash function producing a 64-bit output:
n ≈ √(2 * log(1/(1 - 0.5)))
n ≈ √(2 * log(2))
n ≈ √(2 * 0.693)
n ≈ √(1.386)
n ≈ 1.177
For ε = 0.5, approximately 1.177 random inputs are required to have a probability of collision.
For ε = 0.1:
n ≈ √(2 * log(1/(1 - 0.1)))
n ≈ √(2 * log(10))
n ≈ √(2 * 2.303)
n ≈ √(4.606)
n ≈ 2.147
For ε = 0.1, approximately 2.147 random inputs are needed for a collision.
Similarly, we can calculate the number of inputs required for the hash functions producing outputs of lengths 128 and 160 bits using the same formula.
Please note that these calculations provide approximate values based on the birthday paradox principle. The actual probability of collision may vary depending on the specific characteristics of the hash functions and the nature of the inputs.
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