Sofia invested $360 in an account in the year 2005, and the value has been growing exponentially at a constant rate. The value of the account reached $420 in the year 2009. Determine the value of the account, to the nearest dollar, in the year 2015.
To determine the value of the account in the year 2015, we can use the formula for exponential growth:
\[A = P \times (1 + r)^t\]
Where:
A = final amount
P = initial amount
r = growth rate
t = time in years
Given that Sofia invested $360 in the year 2005 and the value reached $420 in the year 2009, we can use this information to find the growth rate (r).
\[420 = 360 \times (1 + r)^4\]
Now, let's solve for r:
\[1 + r = \left(\frac{420}{360}\right)^\frac{1}{4}\]
\[1 + r = 1.037037\]
\[r = 1.037037 - 1\]
\[r = 0.037037\]
Now that we have the growth rate (r), we can calculate the value of the account in the year 2015 (10 years after 2005):
\[A = 360 \times (1 + 0.037037)^{10}\]
\[A \approx 360 \times 1.439622\]
\[A \approx 518.26\]
Therefore, the value of the account in the year 2015 is approximately $518 (to the nearest dollar).
To determine the value of the account in the year 2015, we can use the exponential growth formula:
V = P * (1 + r)^t
Where:
V = Value of the account in the year we want to find (2015)
P = Initial investment or principal amount ($360)
r = Growth rate (unknown)
t = Number of years since the initial investment
Given that the value of the account reached $420 in 2009 (4 years after the initial investment in 2005), we can substitute these values into the formula:
420 = 360 * (1 + r)^4
Now, we need to solve for r. To do this, we can rearrange the equation:
(1 + r)^4 = 420/360
Next, we take the fourth root of both sides to isolate (1 + r):
1 + r = (420/360)^(1/4)
By subtracting 1 from both sides, we can solve for r:
r = (420/360)^(1/4) - 1
Now that we have the growth rate (r), we can find the value of the account in 2015. This is the number of years after the initial investment (2015 - 2005 = 10 years):
V = 360 * (1 + r)^10
Calculating this, we have:
r = (420/360)^(1/4) - 1 ≈ 0.0361 (rounded to four decimal places)
V = 360 * (1 + 0.0361)^10 ≈ $494
Therefore, the value of the account in the year 2015 is approximately $494.