Question 1 of 18, Step 1 of 1 Growing linearly, the balance owed on your credit card doubles from \( \$ 800 \) to \( \$ 1600 \) in 6 months. If the balance were growing according to the exponential function \( f(x)=800(1+0.122)^{x} \) where \( x \) represents the number of months, what would the balance be after 6 months? Round your answer to the nearest cent. Answer How to enter your answer (opens in new window) Keyboard Shortrus Kist

To determine the balance after 6 months using the exponential growth model \( f(x) = 800(1 + 0.122)^x \), where \( x \) represents the number of months, follow these steps: ### Step 1: Identify the Given Information - **Initial Balance (\( f(0) \))**: \$800 - **Growth Rate per Month**: 12.2% (or 0.122 in decimal form) - **Time (\( x \))**: 6 months ### Step 2: Apply the Exponential Growth Formula The exponential growth formula is: \[ f(x) = 800(1 + 0.122)^x \] Plugging in \( x = 6 \): \[ f(6) = 800(1 + 0.122)^6 = 800(1.122)^6 \] ### Step 3: Calculate \( (1.122)^6 \) Let's compute \( (1.122)^6 \): 1. **First Month:** \[ 1.122^1 = 1.122 \] 2. **Second Month:** \[ 1.122^2 = 1.122 \times 1.122 \approx 1.258884 \] 3. **Third Month:** \[ 1.122^3 = 1.258884 \times 1.122 \approx 1.412477 \] 4. **Fourth Month:** \[ 1.122^4 = 1.412477 \times 1.122 \approx 1.5848 \] 5. **Fifth Month:** \[ 1.122^5 = 1.5848 \times 1.122 \approx 1.7781 \] 6. **Sixth Month:** \[ 1.122^6 = 1.7781 \times 1.122 \approx 1.9951 \] So, \[ (1.122)^6 \approx 1.9951 \] ### Step 4: Calculate the Final Balance Multiply the result by the initial balance: \[ f(6) = 800 \times 1.9951 \approx 1596.08 \] ### Step 5: Round to the Nearest Cent \[ f(6) \approx \$1596.08 \] ### **Answer** After 6 months, the balance would be **\$1,596.08**.