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Sagot :
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.
### 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.
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