To solve this problem, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future amount (15,528.23 in this case)
P = the principal or initial deposit ($100 in this case)
r = the interest rate (0.05 or 5% in this case)
n = the number of times interest is compounded per year (12 for monthly compounding in this case)
t = the number of years (10 in this case)
Plugging in the given values into the formula, we have:
15,528.23 = 100(1 + 0.05/12)^(12*10)
Simplifying the equation further:
155.2823 = (1 + 0.004167)^(120)
Taking the natural logarithm (ln) of both sides to isolate the exponent:
ln(155.2823) = ln[(1 + 0.004167)^(120)]
Using a calculator to compute the natural logarithm on the left side:
ln(155.2823) ≈ 5.043558
Now, divide both sides by 120:
5.043558/120 ≈ 0.042030
This is the value of (1 + 0.004167), so subtract 1 to find the value of r/n:
0.042030 - 1 ≈ -0.957970
Now, divide both sides by 0.004167:
-0.957970/0.004167 ≈ -230.374707
Multiplying both sides by -1, we have:
230.374707 ≈ 1/[(1 + 0.004167)^(12*10)]
Take the reciprocal on the right side:
1/230.374707 ≈ (1 + 0.004167)^(12*10)
This shows that (1 + 0.004167)^(12*10) is approximately equal to 0.004341260.
Subtracting 1, we find that the value of r/n is approximately 0.003341260.
Now, multiply r/n by 12 to find the value of r:
0.003341260 * 12 ≈ 0.040095
Thus, the interest rate r is approximately 0.040095 or 4.01%.
Therefore, 15,528.23 can be expressed as $1.552823 x 10^4, where a = 1.
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 x 10^n where a is single digit
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