$9000 is invested at 8% compounded quarterly. In how many years will the account have 20)
grown to $11,000? Round your answer to the nearest tenth of a year.
We have to solve for n , where n is the number of quarter years
9000(1.02)^n = 11000
1.02^n = 1.222222...
log both sides
log (1.02^n) = log 1.22222..
n log 1.02 = log 1.22222..
n = log 1.22222/log 1.02 = 10.1335 or 10.1 years
To find the number of years it will take for the account to grow to $11,000, we need to use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A is the future value of the investment ($11,000),
P is the principal amount ($9,000),
r is the annual interest rate (8% or 0.08),
n is the number of times interest is compounded per year (quarterly, so 4 times),
t is the number of years.
Plugging in the given values, the formula becomes:
$11,000 = $9,000(1 + 0.08/4)^(4t)
Now, we can solve for t by rearranging the equation and isolating the variable t.
Divide both sides of the equation by $9,000:
$11,000/$9,000 = (1 + 0.08/4)^(4t)
Simplify the equation:
1.2222... = (1.02)^(4t)
Take the logarithm of both sides of the equation to solve for t:
log(1.2222...) = log((1.02)^(4t))
Using logarithmic properties, we can rewrite the equation as:
log(1.2222...) = 4t * log(1.02)
Now, we can solve for t by dividing both sides of the equation by 4 * log(1.02):
t = log(1.2222...) / (4 * log(1.02))
Using a calculator, we can find:
t ≈ 9.1 years
Therefore, it will take approximately 9.1 years for the account to grow to $11,000 when invested at 8% compounded quarterly.