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)
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We can use the formula for compound interest:
A = P(1 + r/n)^(nt)
where:
A = the future value of the investment
P = the principal amount (initial deposit)
r = annual interest rate (in decimal form)
n = number of times the interest is compounded per year
t = number of years
In this case, we have:
A = $15,528.23 (future value)
P = $100 (monthly deposit)
r = 5% = 0.05 (interest rate in decimal form)
n = 12 (compounded monthly)
t = 10 (years)
Plugging in these values, we can solve for A:
15,528.23 = 100(1 + 0.05/12)^(12*10)
Dividing both sides of the equation by 100:
155.2823 = (1 + 0.05/12)^(12*10)
Taking the 10th root of both sides:
(1 + 0.05/12) = (155.2823)^(1/(12*10))
Simplifying the right side:
(1 + 0.05/12) = (155.2823)^(1/120)
Let's calculate the right side:
(1 + 0.05/12) = 1.00373
Now we can express the original amount of $15,528.23 in the form of a×10^n:
a = 1.552823
n = 4
Thus, $15,528.23 can be expressed as 1.552823 × 10^4.
bot $15,528.23 in the form of a×10^n
where a
is single digit.
To find the amount in the bank after 10 years with monthly deposits of $100 and a 5% interest rate compounded monthly, we can use the formula for the future value of an ordinary annuity:
FV = P * (((1 + r)^n) - 1) / r
Where:
FV = Future value
P = Monthly payment
r = Interest rate per period
n = Number of periods
In this case, P = $100, r = 5%/12 = 0.4167% per month, and n = 10 * 12 = 120 months.
Substituting the values into the formula:
FV = 100 * (((1 + 0.4167%)^120) - 1) / 0.4167%
Calculating this expression gives the value of FV as $15,528.23.
To express $15,528.23 in the form a×10^n, we need to determine a and n.
In scientific notation, $15,528.23 can be written as 1.552823 × 10^4.
Thus, the amount $15,528.23 can be expressed as 1.552823 x 10^4.
To solve this problem, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final amount (in this case, $15,528.23)
P = the principal amount (the initial deposit, in this case, $100 monthly)
r = annual interest rate (in decimal form, in this case, 5%, so r = 0.05)
n = number of times the interest is compounded per year (in this case, monthly, so n = 12)
t = number of years (in this case, 10 years)
Plugging in the given values into the formula, we have:
$15,528.23 = 100(1 + 0.05/12)^(12*10)
Now, we need to solve for (1 + 0.05/12)^(12*10). To do this, we can isolate the exponential term and take the natural logarithm of both sides of the equation:
ln($15,528.23) = ln[100(1 + 0.05/12)^(12*10)]
Using the properties of logarithms, we can simplify the equation further:
ln($15,528.23) = ln(100) + ln[(1 + 0.05/12)^(12*10)]
Now, we can calculate the right-hand side of the equation:
ln($15,528.23) = ln(100) + ln[(1 + 0.05/12)^(120)]
Using a calculator, we can evaluate ln($15,528.23) and ln(100), and then subtract ln(100) from both sides to isolate ln[(1 + 0.05/12)^(120)].
After solving for ln[(1 + 0.05/12)^(120)], we can take the antilogarithm (exponentiating both sides) to find the value of (1 + 0.05/12)^(120).
Finally, to express the amount $15,528.23 in the form of a×10^n, we can convert it to scientific notation. We move the decimal point until we have one non-zero digit to the left of the decimal point, and count the number of places we moved the decimal point to get the exponent.
For example, if the scientific notation of $15,528.23 is 1.552823 × 10^4, then a = 1.552823 and n = 4.