Jayden invested dollar sign, 1, comma, 500$1,500 in an account in the year 2005, and the value has been growing exponentially at a constant rate. The value of the account reached dollar sign, 1, comma, 800$1,800 in the year 2011. Determine the value of the account, to the nearest dollar, in the year 2019.

To solve this problem, we can model the account balance growth using the exponential growth formula:

\[
A(t) = A_0 \cdot e^{kt}
\]

Where:
- $ A(t) $ is the amount of money in the account at time $ t $,
- $ A_0 $ is the initial amount (the value in 2005),
- $ k $ is the exponential growth rate,
- $ t $ is the time elapsed (in years),
- $ e $ is the base of the natural logarithm.

### Step 1: Determine the rate of growth $ k $
We know the following:
- The initial investment, $ A_0 = 1,500 $ dollars in 2005.
- The value in 2011, $ A(2011) = 1,800 $ dollars.

To find $ k $, we can use the formula with $ t = 2011 - 2005 = 6 $ years:

\[
1,800 = 1,500 \cdot e^{6k}
\]

Divide both sides by 1,500:

\[
\frac{1,800}{1,500} = e^{6k}
\]

\[
1.2 = e^{6k}
\]

Now, take the natural logarithm (ln) of both sides to solve for $ k $:

\[
\ln(1.2) = 6k
\]

\[
k = \frac{\ln(1.2)}{6}
\]

Using a calculator:

\[
\ln(1.2) \approx 0.1823
\]

\[
k \approx \frac{0.1823}{6} \approx 0.03038
\]

### Step 2: Find the value of the account in 2019
Now that we know $ k $, we can use the formula to find the value of the account in 2019. The time elapsed between 2005 and 2019 is $ t = 2019 - 2005 = 14 $ years.

\[
A(2019) = 1,500 \cdot e^{0.03038 \times 14}
\]

First, calculate the exponent:

\[
0.03038 \times 14 \approx 0.4253
\]

Now, calculate the value of the account:

\[
A(2019) = 1,500 \cdot e^{0.4253}
\]

\[
A(2019) \approx 1,500 \cdot 1.5301 \approx 2,295
\]

### Final Answer:
The value of the account in 2019 is approximately **$2,295**.

Would you like more details or further clarification on any step?

### Related Questions:
1. How would the account value change if the initial investment was doubled?
2. What would happen if the growth rate $ k $ were reduced to half of its current value?
3. How can we use the compound interest formula to model the same problem?
4. How would you graph the exponential growth of this account over time?
5. What is the effect of compounding frequency on exponential growth?

**Tip**: When solving exponential growth problems, always make sure to isolate the variable (in this case, the growth rate $ k $) first before applying logarithms.

Jayden invested $1,500 in an account in the year 2005, and the value has been growing exponentially at a constant rate. The value of the account reached $1,800 in the year 2011. Determine the value of the account, to the nearest dollar, in the year 2019.

Learn how to calculate the future value of an account growing exponentially. Jayden's initial investment of $1,500 in 2005 grew to $1,800 by 2011. This step-by-step solution shows how to determine the value in 2019, using the exponential growth formula and natural logarithms.

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