It takes 10 years for a $100 monthly bank deposit to amount to $15,528 23 when invested at 5% Interest compounded monthly. Round to the nearest place value, then express the amount $15,528 23 in the form of a * 10 ^ n where a is single digit. (1 point)
To solve this problem, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
where:
A = the future amount
P = the principal amount (initial deposit)
r = annual interest rate (expressed as a decimal)
n = number of times interest is compounded per year
t = number of years
Given:
P = $100
r = 5% = 0.05
n = 12 (compounded monthly)
t = 10 years
Substituting the given values into the formula, we get:
15,528.23 = 100(1 + 0.05/12)^(12*10)
Next, let's solve for (1 + 0.05/12)^(12*10):
15,528.23/100 = (1 + 0.05/12)^(12*10)
155.2823 = (1.0041667)^(120)
Taking the natural logarithm of both sides:
ln(155.2823) = ln[(1.0041667)^120]
ln(155.2823) = 120 * ln(1.0041667)
Using a calculator, we find:
ln(155.2823) = 120 * 0.004162942
ln(155.2823) ≈ 0.4990343144
Now, we have:
0.4990343144 = ln[(1 + 0.05/12)^(12*10)]
To solve for (1 + 0.05/12)^(12*10), we need to take the inverse natural logarithm (e^x) of both sides:
(1 + 0.05/12)^(12*10) = e^0.4990343144
Using a calculator, we find:
(1 + 0.05/12)^(12*10) ≈ e^0.4990343144 ≈ 1.64843
Therefore, the amount after 10 years is approximately $1.64843 * 100 = $164.843
To express $15,528.23 in the form of a * 10 ^ n, we can write it as 1.64843 * 10^4.