a university alumni group wishes to provide an annual scholarship in the amount of 1500 beginning next year. if the scholarship fund will earn an interest rate of 8%/year compounded continuously, find the amount of the endowment the alumni are required to make now.
To calculate the amount of the endowment the alumni group needs to make, we can use the formula for continuous compound interest:
A = P * e^(rt)
Where:
A = Final amount (future value)
P = Principal (initial amount)
r = Annual interest rate (as a decimal)
t = Time period in years
e = Euler's number (approximately 2.71828)
In this case, the final amount (A) is the scholarship amount of $1500, the interest rate (r) is 8% or 0.08, and the time period (t) is 1 year. We need to find the principal amount (P) or the endowment.
Let's substitute the values into the formula and solve for P:
1500 = P * e^(0.08*1)
Now, divide both sides of the equation by e^(0.08):
1500 / e^(0.08) = P
Using a calculator, we find that e^(0.08) is approximately 1.08328706767. Let's plug this value into the equation:
1500 / 1.08328706767 = P
After performing the division, we find that:
P ≈ $1385.44
Therefore, the alumni group should make an endowment of approximately $1385.44 in order to provide an annual scholarship of $1500.
To find the amount of the endowment the alumni are required to make, we need to use the formula for continuous compounding:
A = P * e^(rt)
Where:
A = The future value (endowment)
P = The principal amount (required endowment)
e = The mathematical constant approximately equal to 2.71828
r = The interest rate per year (8% or 0.08)
t = The time period in years (1 year)
We can rearrange the formula to solve for P:
P = A / e^(rt)
Given that the future value (A) is $1,500 and the interest rate (r) is 8% or 0.08, we can calculate the required endowment (P).
P = 1500 / e^(0.08*1)
To find the value of e^(0.08*1), we can use a scientific calculator or an online calculator. The value is approximately 1.083287.
P = 1500 / 1.083287
By dividing $1,500 by 1.083287, we find that the required endowment the alumni are required to make is approximately $1,385.84.
Therefore, the alumni need to make an endowment of approximately $1,385.84 now to provide an annual scholarship of $1,500 starting next year, assuming an interest rate of 8% compounded continuously.
1500 = P(e^.08-1)
P = 18010