Melanie puts $1,100 in an investment account that she expects will make 5% interests for each three month period. However, after a year she realizes she was wrong about the interest rate and she has $50 less than she expected. Assuming the interest rate the account earns is constant, which of the following equations expresses the total money ,x, she will after t years using the actual rate?

Options

A. `x=1,100(1.04)^(4t)`
B. `x=1,100(1.05)^(4t-50)`
C. `x=1,100(1.04)^(t//3)`
D. `x=1,100(1.035)^(4t)`

Answer

To solve the problem, we need to express the total amount of money Melanie will have after \( t \) years using the actual interest rate. Here’s a step-by-step solution:

### Step 1: Calculate the Expected Amount After One Year
Melanie initially assumes a 5% interest rate for each 3-month period. Since there are 4 quarters in a year, the total amount after one year using the assumed interest rate can be calculated as follows:

\[ A = P \left(1 + r\right)^n \]

Where:
- \( P = 1100 \) (the principal amount)
- \( r = 0.05 \) (the interest rate per period)
- \( n = 4 \) (the number of periods in one year)

Substituting the values:

\[ A = 1100 \left(1 + 0.05\right)^4 \]

### Step 2: Calculate the Amount
Now we calculate the amount:

\[ A = 1100 \left(1.05\right)^4 \]

Calculating \( (1.05)^4 \):

\[ (1.05)^4 \approx 1.21550625 \]

Now, substituting back:

\[ A \approx 1100 \times 1.21550625 \approx 1337.06 \]

### Step 3: Determine the Actual Amount
Melanie realizes she has $50 less than expected after one year. Therefore, the actual amount she has is:

\[ A_{actual} = A - 50 = 1337.06 - 50 = 1287.06 \]

### Step 4: Set Up the Equation for Actual Rate
Let \( R \) be the actual interest rate per period. The equation for the actual amount after one year is:

\[ A_{actual} = P \left(1 + \frac{R}{100}\right)^n \]

Substituting the known values:

\[ 1287.06 = 1100 \left(1 + \frac{R}{100}\right)^4 \]

### Step 5: Solve for \( R \)
First, divide both sides by 1100:

\[ \frac{1287.06}{1100} = \left(1 + \frac{R}{100}\right)^4 \]

Calculating the left side:

\[ \frac{1287.06}{1100} \approx 1.17196 \]

Now we have:

\[ 1.17196 = \left(1 + \frac{R}{100}\right)^4 \]

### Step 6: Take the Fourth Root
Taking the fourth root of both sides:

\[ 1 + \frac{R}{100} = (1.17196)^{1/4} \]

Calculating the fourth root:

\[ (1.17196)^{1/4} \approx 1.04 \]

### Step 7: Solve for \( R \)
Subtract 1 from both sides:

\[ \frac{R}{100} = 0.04 \]

Multiplying by 100 gives:

\[ R = 4 \]

### Step 8: Write the Final Equation for Total Amount After \( t \) Years
Now we can express the total amount \( x \) after \( t \) years using the actual interest rate:

\[ x = 1100 \left(1 + \frac{4}{100}\right)^{4t} \]

This simplifies to:

\[ x = 1100 \left(1.04\right)^{4t} \]

### Final Answer
The equation that expresses the total money \( x \) she will have after \( t \) years using the actual rate is:

\[ x = 1100 \times (1.04)^{4t} \]

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