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To solve this problem, let's break it down step by step:

### 1. **Writing the Exponential Equation:**

The general form of an exponential growth equation is:
\[
y = a(b)^x
\]
where:
- $ a $ is the initial amount,
- $ b $ is the growth factor (rate of increase),
- $ x $ is the number of years,
- $ y $ is the amount of money in the account after $ x $ years.

**Given:**
- Patrick's initial deposit is $ a = 894 $ dollars.
- The account grows by **2%** per year. This means the growth factor $ b = 1 + 0.02 = 1.02 $.

Thus, the equation becomes:
\[
y = 894(1.02)^x
\]
This is the exponential equation that models the amount of money in Patrick's account.

### 2. **Finding When the Account Exceeds $1,000:**

We need to find out after how many years $ x $ the account will have more than $1,000. This means we want to solve the equation:
\[
1000 = 894(1.02)^x
\]

To solve for $ x $:
\[
\frac{1000}{894} = (1.02)^x
\]
\[
1.1189 = (1.02)^x
\]

Now, take the natural logarithm (ln) of both sides:
\[
\ln(1.1189) = x \ln(1.02)
\]
\[
x = \frac{\ln(1.1189)}{\ln(1.02)}
\]
Using a calculator:
\[
x \approx \frac{0.11241}{0.01980} \approx 5.68
\]

So, Patrick's account will have more than $1,000 after about **6 years**.

---

Do you have any questions or would you like more details? Here are some related questions to extend your understanding:

1. What would the balance be after 10 years?
2. How would the equation change if the interest rate was 3% instead of 2%?
3. How does exponential growth differ from simple interest growth?
4. Can you graph the function $ y = 894(1.02)^x $ to visualize the growth?
5. What happens if Patrick adds more deposits each year? How would that affect the equation?

**Tip:** When solving exponential equations, using logarithms is a powerful tool for isolating the exponent.

Patrick spent last summer working in construction with his dad, and he put $894 of his earnings into a new high-yield savings account. The money in this savings account will increase by 2% each year. Write an exponential equation in the form y = a(b)^x that can model the amount of money in Patrick's account, y, x years after starting the account. If Patrick makes no other deposits or withdrawals, after how many years will his account have more than $1,000?

Learn how to model exponential growth for Patrick's savings account, which starts with $894 and grows by 2% annually. This solution demonstrates how to set up the exponential equation y = 894(1.02)^x and calculate when the account balance will exceed $1,000. Perfect for Grades 9-12 students exploring exponential functions and logarithms.

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