Final answer:
To the nearest dollar, Kayden would have approximately $1482 in his account when Ian's money has tripled in value.
Step-by-step explanation:
To find out how much money Kayden will have in his account when Ian's money has tripled in value, we need to calculate the future value of Ian's account.
Using the formula for compound interest, the future value, FV, is given by:
FV = P(1 + r/n)^(nt)
Where:
P = principal amount
r = annual interest rate (as a decimal)
n = number of times interest is compounded per year
t = number of years
In this case, Ian's principal amount is $530, the interest rate is 6 1/2% or 0.065, and the interest is compounded continuously.
Thus, we can use the formula:
FV = 530 * e^(0.065 * t)
Since we want to find out when Ian's money has tripled, we set FV equal to 3 times the principal amount and solve for t:
3 * 530 = 530 * e^(0.065 * t)
Simplifying the equation, we have:
3 = e^(0.065 * t)
Taking the natural logarithm of both sides, we get:
ln(3) = 0.065 * t
Dividing both sides by 0.065, we find:
t = ln(3) / 0.065
Using a calculator, we can approximate t to be approximately 15.35 years.
Now that we know the time required for Ian's money to triple, we can calculate the future value of Kayden's account after the same time period.
Using the formula for compound interest, we have:
FV = 530 * (1 + 6 7/8%)^15.35
Using a calculator to find the future value of Kayden's account, we get:
FV ≈ $1482
Therefore, to the nearest dollar, Kayden would have approximately $1482 in his account when Ian's money has tripled in value.