Question* Problem 2
Groundwater well is known to begin pumping sand once it becomes exploited (old), and this may damage the subsequent water treatment processes. To solve this problem, two alternatives are proposed:
- A new well can be drilled at a capital cost of with minimal operating and maintenance expenses of per year.
- A settling tank can be constructed ahead of the treatment processes which will cost to build and per year to operate and maintain.
The salvage value of either option at EOY 20 is of the capital investment. Using a MARR of .
Studdy Solution
STEP 1
What is this asking?
Which is cheaper over 20 years: a new well or a settling tank?
Watch out!
Don't forget to include the salvage value!
Also, remember that costs are *negative* cash flows.
STEP 2
1. Calculate the present worth of the new well.
2. Calculate the present worth of the settling tank.
3. Compare the present worths.
STEP 3
Alright, let's **crunch some numbers** for this new well!
The **initial cost** is , so that's a negative cash flow right now: .
STEP 4
We've got **annual maintenance costs** of , and we need to find the present worth of these costs over **20 years**.
The formula for the present worth of a uniform series is:
Where is the **annual cost**, is the **interest rate**, and is the **number of years**.
STEP 5
Plugging in our **values**, we get: So the **present worth** of the maintenance costs is approximately (remember, costs are negative!).
STEP 6
Now, let's **factor in the salvage value**!
It's of the **initial investment**, so that's .
We need to find the **present worth** of this amount received 20 years from now.
The formula for present worth of a single future amount is:
STEP 7
Plugging in our **values**, we get: That's a positive cash flow, so .
STEP 8
Finally, let's **add everything up** to get the **total present worth** of the new well:
STEP 9
The **initial cost** of the settling tank is .
STEP 10
The **annual maintenance** is .
Using the same present worth formula as before, with and , we get:
So the present worth of the maintenance is approximately .
STEP 11
The **salvage value** is .
Its present worth is:
STEP 12
The **total present worth** of the settling tank is:
STEP 13
The **present worth** of the new well is .
The **present worth** of the settling tank is .
Since we want to *minimize* cost (which means maximizing the present worth, since costs are negative), the new well is the better option.
STEP 14
The new well is the more economical option, with a present worth of compared to the settling tank's .
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