How much money has to be invested at 4.3% interest compounded continuously to have $19,000 after 16 years?
x e^.043(16) = 19000
x = 19000/e^(.688)
= 9549.02
Pe^(.043*16) = 19000
P = 9549.02
To determine the amount of money that needs to be invested at 4.3% interest compounded continuously to reach $19,000 after 16 years, we can use the formula for continuous compound interest:
A = P * e^(rt)
Where:
A = Final amount ($19,000 in this case)
P = Principal amount (the initial investment)
e = Euler's number (approximately 2.71828)
r = Annual interest rate (4.3% or 0.043 as a decimal)
t = Time (16 years)
We need to isolate the principal amount (P) in order to solve for it. Rearranging the formula, we have:
P = A / (e^(rt))
Substituting the given values:
P = 19,000 / (e^(0.043 * 16))
To calculate this, we need the value of e raised to the power of (0.043 * 16). Let's do the calculations step by step:
Step 1: Calculate the exponent
Exponent = 0.043 * 16 = 0.688
Step 2: Calculate e raised to the exponent
e^(0.688) ≈ 1.990
Step 3: Calculate the principal amount
P = 19,000 / 1.990 ≈ $9,547.74
Therefore, approximately $9,547.74 needs to be invested at 4.3% interest compounded continuously to accumulate $19,000 after 16 years.