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×10n
where a
is single digit.(1 point)
To solve this problem, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment
P = the amount of the monthly deposit ($100)
r = the annual interest rate (5% or 0.05)
n = the number of times interest is compounded per year (12 times monthly)
t = the number of years (10 years)
Plugging in the values, we get:
$15,528.23 = $100(1 + 0.05/12)^(12*10)
Dividing both sides by $100:
155.2823 = (1 + 0.05/12)^(12*10)
Taking the natural logarithm of both sides:
ln(155.2823) = ln((1 + 0.05/12)^(12*10))
Using logarithm properties, we can bring down the exponent:
ln(155.2823) = (12*10) * ln(1 + 0.05/12)
Dividing both sides by 120:
ln(155.2823)/120 = ln(1 + 0.05/12)
Rounding to the nearest place value, we have:
ln(155.28)/120 = 0.003973
Now, multiplying both sides by 120:
120 * 0.003973 = ln(1 + 0.05/12)
0.4776 = ln(1 + 0.05/12)
Using the inverse natural logarithm (exponential function), we compute:
e^0.4776 = 1 + 0.05/12
1.6126 ≈ 1 + 0.00417
Finally, in the form a × 10^n, we have:
1.6126 ≈ 1 × 10^0+0.00417
Rounding to the nearest place value, we get:
1.6126 ≈ 1.612 × 10^0
To find the interest rate compounded monthly, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final amount ($15,528.23)
P = the initial deposit ($100)
r = the interest rate (unknown)
n = the number of times interest is compounded per year (12, since it is compounded monthly)
t = the number of years (10)
Plugging in the given values, we have:
$15,528.23 = $100(1 + r/12)^(12*10)
Divide both sides of the equation by $100:
155.2823 = (1 + r/12)^(120)
Taking the 120th root of both sides:
(1 + r/12) = 155.2823^(1/120)
Subtracting 1 from both sides:
r/12 = 155.2823^(1/120) - 1
Multiply both sides by 12:
r = 12 * (155.2823^(1/120) - 1)
Calculating this value, we find that r ≈ 0.3950.
Therefore, the interest rate is approximately 0.3950 or 39.50% when rounded to the nearest place value.
Next, we need to express $15,528.23 in the form of a × 10^n, where a is a single digit.
Since $15,528.23 is greater than 10, we can write it as:
15,528.23 = 1.552823 × 10^4
To solve this problem, we need to use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final amount (15,528.23)
P = the principal amount (the amount being deposited each month) ($100)
r = the annual interest rate (5% or 0.05)
n = the number of times interest is compounded per year (12, since it's compounded monthly)
t = the number of years (10)
Now we can substitute the given values into the formula:
15,528.23 = 100(1 + 0.05/12)^(12*10)
To simplify, divide 0.05 by 12:
15,528.23 = 100(1 + 0.00417)^(12*10)
Now calculate the exponent:
15,528.23 = 100(1.00417)^(120)
Use a calculator to evaluate:
15,528.23 ≈ 100(1.67472)
15,528.23 ≈ 167.472
Now, round to the nearest place value:
15,528.23 ≈ 167.47
Finally, we can express this value in the form of a × 10^n. Let's rearrange:
167.47 = 1.6747 × 10^2
Therefore, the amount $15,528.23 can be expressed as approximately 1.6747 × 10^2.