A company invests $15,000.00 in an account that compounds interest annually. After two years, the account is worth $16,099.44. Use the function, where r is the annual interest rate, P is the principal, and A is the amount of money after t years. What is the interest rate of the account?
(1 point)
Responses
1.04%
1.04%
3.6%
3.6%
5.4%
5.4%
7.3%
7.3%
To find the interest rate, we can use the formula A = P(1 + r)^t and plug in the given values.
$16,099.44 = $15,000(1 + r)^2
$16,099.44 = $15,000(1 + r)(1 + r)
$16,099.44 = $15,000(1 + 2r + r^2)
$16,099.44 = $15,000 + $30,000r + $15,000r^2
Now, we can solve for r by rearranging the terms:
$15,099.44 = $30,000r + $15,000r^2
$15,099.44 = $15,000r^2 + $30,000r
$1,099.44 = $15,000r^2 + $30,000r
Divide through by $15,000:
0.073296 = r^2 + 2r
Now, we need to solve for r. We can use the quadratic formula to find the interest rate:
r = (-2 ± sqrt(2^2 - 4(1)(0.073296))) / 2(1)
r = (-2 ± sqrt(4 - 0.293184)) / 2
r = (-2 ± sqrt(3.706816)) / 2
r = (-2 ± 1.9243934559) / 2
We take the positive value:
r = (-2 + 1.9243934559) / 2
r = (-0.0756065441) / 2
r = -0.037803272
Converting to a percentage, the interest rate is approximately 3.78%.
Therefore, the closest option provided is 3.6%.