PLS HELP, BEEN STUCK FOREVER ON TWO QUESTIONS, THIS IS ONE :
There are usually no costs for the first 3 years, but thereafter maintenance is re- quired for restriping, weed control, light replacement, shoulder repairs, etc. For one section of a particular highway, these costs are projected to be $6000 in year 3, $7000 in year 4, and amounts increasing by $1000 per year through the highway's expected 30-year life. Assuming it is replaced with a similar roadway, what is its perpetual equivalent annual worth (in years] through infinity) at an interest rate of8% per year?
THIS IS THE OTHER ONE:
7. A permanent endowment At a university is to award scholarships to engineering students. The awards are to be made beginning 6 years after the $12 million lump-sum donation is made. If the interest from the endowment is to fund 100 students each year in the amount of $12,000 each, what annual rate of return must the endowment fund earn?
I HAVE DONE WORK ON THEM, THEY ARE JUST REALLY LONG TO TYPE UP...AND THEY ARE WRONG. PLs help!!! i really need to understand this, any help is appreciated
Sure, I'd be happy to help you with these two questions. Let's start with the first one:
1. Perpetual Equivalent Annual Worth Calculation:
To calculate the perpetual equivalent annual worth, we need to find the equivalent annual maintenance cost that would meet the total cost projection for the highway over its expected 30-year life.
Given:
- Maintenance costs are projected to be $6000 in year 3, $7000 in year 4, and increasing by $1000 per year.
- Interest rate is 8% per year.
Here's how you can calculate it step by step:
Step 1: Find the present value of costs in years 3 to 30.
- We have a constant increase of $1000 per year, so this can be considered an arithmetic series.
- Use the formula for the sum of an arithmetic series: S = (n/2)(a + l), where:
- S is the sum of the series,
- n is the number of terms in the series (30 - 4 + 1 = 27 terms),
- a is the first term of the series ($7000), and
- l is the last term of the series ($7000 + $1000 * (27 - 1) = $7000 + $26,000 = $33,000).
- Calculate the sum S = (27/2)(7000 + 33000) = 27 * 20000 = $540,000.
Step 2: Calculate the equivalent annual worth.
- We can use the formula for the equivalent annual worth (AE) of a present sum: AE = A * (P/A,i%,n), where:
- AE is the equivalent annual worth,
- A is the present sum ($6000 + $7000 = $13,000 for years 3 and 4),
- P/A,i%,n is the present worth factor for the interest rate and number of years.
- Find the present worth factor for 8% interest and 27 years: P/A,8%,27 = 1 / (1 + 0.08)^27 = 0.0796 (rounded to four decimal places).
- Calculate the equivalent annual worth AE = $13,000 * 0.0796 = $1,034.80 (rounded to two decimal places).
Therefore, the perpetual equivalent annual worth of the maintenance costs is approximately $1,034.80.
Now, let's move on to the second question:
2. Annual Rate of Return Calculation:
To find the annual rate of return the endowment fund must earn, we need to calculate the interest rate that will generate enough annual interest to fund 100 students per year at $12,000 each, starting from 6 years after the donation is made.
Given:
- Lump-sum donation: $12 million.
- Scholarship amount per year: $12,000 for 100 students.
- Funding for scholarships starts 6 years after the donation.
Here's how you can calculate it step by step:
Step 1: Find the total funding required for scholarships.
- The scholarship amount per year for 100 students is $12,000, so the total funding required per year is $12,000 * 100 = $1,200,000.
Step 2: Find the future value of the lump-sum donation after 6 years.
- Use the compound interest formula: FV = PV * (1 + r)^n, where:
- FV is the future value,
- PV is the present value ($12 million),
- r is the interest rate, and
- n is the number of years (6 years).
- Rearrange the formula to solve for the interest rate: r = (FV / PV)^(1/n) - 1.
- Calculate the future value: FV = $12 million * (1 + r)^6.
- Divide the total funding required ($1,200,000) by the future value to get the interest rate: r = (1,200,000 / (12 million * (1 + r)^6))^(1/6) - 1.
The equation is now a bit complex, and it might be best to solve it using numerical methods like trial and error or using a spreadsheet software. Start with an assumption for the interest rate, calculate the left side of the equation, and adjust the interest rate until the equation is satisfied (the left side is very close to 1).
Therefore, to find the specific interest rate that satisfies the equation, you can use trial and error or a numerical method in a spreadsheet program like Microsoft Excel or Google Sheets.
I hope this explanation helps you understand the process of solving these two questions.