At the time of the birth of a child, a parent wants to begin a college fund that will grow to $50000 by the child's 18th birthday. Interest is compounded continuously at 8.5%. What should the initial investment (P0) be?
how many years will it take for $4,000 to reach $6,000 at a simple interest rate of 5
To find the initial investment (P0) required to grow to $50,000 by the child's 18th birthday, we can use the formula for continuous compound interest:
A = P*e^(rt)
Where:
A = the final amount (in this case, $50,000)
P = the principal amount (initial investment)
e = Euler's number (approximately 2.71828)
r = annual interest rate (8.5% or 0.085)
t = time in years (18 years)
Substituting the given values into the formula, we have:
$50,000 = P * e^(0.085 * 18)
To find P0, we can rearrange the equation:
P0 = $50,000 / e^(0.085 * 18)
Calculating P0:
P0 = $50,000 / 2.71828^(0.085 * 18)
P0 ≈ $11,146.17
Therefore, the initial investment (P0) should be approximately $11,146.17.
To determine the initial investment (P0) needed to grow to $50,000 by the child's 18th birthday with continuous compounding interest at 8.5%, you can use the formula for compound interest:
P(t) = P0 * e^(rt),
where:
- P(t) is the future value (or target amount) of the investment
- P0 is the initial investment
- e is the mathematical constant approximately equal to 2.71828
- r is the interest rate per time period
- t is the time in years
In this case, P(t) = $50,000, r = 0.085 (8.5% expressed as a decimal), and t = 18 years.
We want to solve for P0, so we can rearrange the formula:
P0 = P(t) / e^(rt)
Now we can plug in the values and calculate:
P0 = $50,000 / e^(0.085 * 18)
Calculating the exponent first: e^(0.085 * 18) ≈ 3.36628
P0 = $50,000 / 3.36628
P0 ≈ $14,840
Therefore, the initial investment (P0) should be approximately $14,840 to grow to $50,000 by the child's 18th birthday with continuous compounding interest at 8.5%.
50,000=PO*e^k18
ln 50,000= ln PO + .085*18
lnPO= 10.82-1.53=9.29
PO=10829.18
check that.