A wealthy patron of a small private college wishes to endow a chair in mathematics with a gift of G thousand dollars. Suppose the mathematician who occupies the chair is to receive $110 thousand dollars per year in salary and benefits. If money costs 8% per year compounded continuously, what is the smallest possible value for G?
Since it's compunded continuously, you use Present Value =
Integrate from 0-T f(t)e^(-rt).
It doesn't give a time limit, so we'll assume it's infinte.
Integrate from 0-Infinity 110e^(-.08t)
=[-1375e^-.08(Inifinity)]-[-135e^-.08(0)]
= 0-(-1375)
= 1375
So they will need as much in the account to annually generate 10,000 in interest
x e^.08 - x = 10000
x(e^.08 - 1) = 10000
x = 10000/(e^.08 - 1) = $120,066.66
Since G is supposed to in thousands
G ≥ 120.06666
check:
amount of the 120066.66 after 1 year of continuous growth of 8%
= 120066.66 e^.08 = 130066.66
amount earned for math guy = 130066.66-120066.6
= 10,000
how did u get 10000
To determine the smallest possible value for G, let's break down the problem step by step:
Step 1: Understand the problem.
We need to find the smallest value for G (in thousands of dollars) that would allow a wealthy patron to endow a chair in mathematics and pay a mathematician $110 thousand per year.
Step 2: Formulate the equation.
We can set up an equation that involves the present value of G. Since the patron wants to pay $110 thousand per year, the equation will include the present value of an annuity formula.
Step 3: Use the present value of an annuity formula.
The present value of an annuity formula is given by:
PV = A / (1 + r)^n
where PV is the present value, A is the future payment, r is the interest rate, and n is the number of periods.
In this case, A (the future payment) is $110 thousand per year, and the interest rate (r) is 8% per year. However, the interest is compounded continuously, so we can convert the interest rate into a continuous rate by using the formula:
r_continuous = ln(1 + r/n)
where n is the number of compounding periods per year.
Since the interest is compounded continuously, n is extremely large or approaches infinity. Thus, we can rewrite the formula as:
r_continuous = ln(1 + r/∞) = ln(1 + 0) = 0
Therefore, the continuous interest rate (r_continuous) is 0.
Step 4: Calculate the present value.
Using the present value of an annuity formula, we can write the equation:
G = A / e^(r_continuous * n)
where G is the present value we want to find, and e is Euler's number (approximately 2.71828).
Since r_continuous = 0, the equation simplifies to:
G = A / e^(0 * n) = A / e^0 = A
Therefore, the smallest value for G is equal to the annual payment, which is $110 thousand.
Step 5: Calculate the smallest possible value for G.
Since G is given in thousands of dollars, we need to divide the annual payment by 1,000:
G = $110,000 / 1,000 = G = 110
So, the smallest possible value for G is 110 thousand dollars.
In summary, the smallest possible value for G is $110,000 (110 in thousands of dollars).