Respuesta :
Answer:
13.2 years
Step-by-step explanation:
To find out how long it would take for the account balance to reach $10,000, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final account balance ($10,000)
P = the initial investment ($2,000)
r = the annual interest rate (4% or 0.04)
n = the number of times interest is compounded per year (quarterly, so n = 4)
t = the number of years we want to find
Let's substitute these values into the formula:
$10,000 = $2,000(1 + 0.04/4)^(4t)
Now, let's solve for t.
Divide both sides of the equation by $2,000:
5 = (1 + 0.01)^(4t)
Take the logarithm of both sides (base doesn't matter):
log(5) = log[(1 + 0.01)^(4t)]
Use the power rule of logarithms to bring the exponent down:
log(5) = 4t * log(1 + 0.01)
Divide both sides of the equation by 4 * log(1 + 0.01):
t = log(5) / (4 * log(1 + 0.01))
Using a calculator, we find that t is approximately 13.2 years (rounded to the nearest tenth).
Therefore, it would take approximately 13.2 years for the account balance to reach $10,000 when $2,000 is invested at 4% annual interest compounded quarterly.
