Final answer:
By using the formula for compound interest and solving for the time variable, we find that it will take approximately 8 years for Madeline's investment to grow to $1,250, with the closest answer being 8 years (option C).
Step-by-step explanation:
To determine how long it will take for Madeline's investment to grow to $1,250 with an interest rate compounded daily, we can use the formula for compound interest:
A = P(1 + \frac{r}{n})^{nt}
Where:
- A is the amount of money accumulated after n years, including interest.
- P is the principal amount (the initial amount of money).
- r is the annual interest rate (decimal).
- n is the number of times that interest is compounded per year.
- t is the time the money is invested for in years.
We need to solve for t, the time in years, and we have the following information:
- A = $1,250
- P = $960
- r = 3.5% or 0.035 (as a decimal)
- n = 365 (since the interest is compounded daily)
Substituting these values into the formula, we get:
$1,250 = $960(1 + \frac{0.035}{365})^{365t}
Now solving for t, we find:
$1,250 / $960 = (1 + \frac{0.035}{365})^{365t}
1.30208333 = (1 + 0.00009589041)^{365t}
Using logarithms to solve for t, we have:
log(1.30208333) = log((1 + 0.00009589041)^{365t})
log(1.30208333) = 365t * log(1 + 0.00009589041)
t = \frac{log(1.30208333)}{365 * log(1 + 0.00009589041)}
After calculating, we find that t is approximately 8.04 years.
Therefore, the closest answer to the nearest year is 8 years, which is option C.