Let's solve the problem step by step in detail.
### Step 1: Understanding the problem
Haley invested [tex]$750 into a mutual fund that pays 3.5% interest compounded annually. We need to find the value of her investment after 15 years.
### Step 2: Identifying variables
- Principal (initial investment) \(P\) = 750
- Annual interest rate \(r\) = 3.5\%
- Number of years \(t\) = 15
### Step 3: Establish the formula
The formula for the compound interest when compounded annually is:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]
Here, \(A\) is the amount after \(t\) years.
### Step 4: Simplified exponential form
The compound interest formula can be converted to match the exponential form \( y = a \cdot b^x \), where:
- \(a\) is the initial amount
- \(b\) is the growth factor
- \(x\) represents the number of times the interest is applied (in this case, years)
So, we can rewrite as:
\[ A = P \cdot (1 + \frac{r}{100})^t \]
\[ y = 750 \cdot (1.035)^x \]
### Step 5: Filling the variables
Now, match the variables to create the equation in the form \( y = a(b)^x \):
- \(a = 750\)
- \(b = 1.035\) (since \(1 + 0.035 = 1.035\))
- \(x = t = 15\) (since we want the value in 15 years)
The equation becomes:
\[ y = 750(1.035)^{15} \]
### Step 6: Solving for y
To find the value of the mutual fund after 15 years, we calculate:
\[ y = 750(1.035)^{15} \]
From the calculations, we know that:
\[ y \approx 1256.51 \]
### Step 7: Answering the questions
1. Equation form:
\[ y = 750(1.035)^x \]
2. What number will you fill in for \(x\) to solve the equation?
To solve the equation, we fill in \(x = 15\).
3. Value of the mutual fund after 15 years (\(y\)):
\[ y \approx \$[/tex]1256.51 \]
So, the value of Haley's mutual fund after 15 years would be approximately $1256.51.
