Mary Invested $$ \$ 50,000 $$ in a retirement account that is increasing in value by $$ 4.5 \% $$ each year. She plans to leave the money there until she retires in 30 years. a. What are the Independent Variable (IV) and Dependent Variable (DV) in this situation? Use a smart label, describe each and include units. b. Using your variables from part a, Write a function equation that models the situation. c. How much money will Mary have in the account after 20 years? d. What is the situtational domain and range for this function? Be sure to include units in your answers.

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**a. Independent and Dependent Variables**  
- **Independent Variable (IV):**  
  Label: $$ t $$  
  Description: Time in years since Mary made her investment.  
  Units: Years

- **Dependent Variable (DV):**  
  Label: $$ A(t) $$  
  Description: The value of the retirement account at time $$ t $$.  
  Units: Dollars

---

**b. Function Equation Modeling the Situation**

Mary’s account grows at a rate of $$ 4.5\% $$ per year. The function that models exponential growth is given by:

$$
A(t) = P(1 + r)^t
$$

Substituting in the values:  
- $$ P = 50000 $$ (initial investment)  
- $$ r = 0.045 $$ (annual growth rate)

The function becomes:

$$
A(t) = 50000 \times (1.045)^t
$$

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**c. Account Value After 20 Years**

We substitute $$ t = 20 $$ into the function:

$$
A(20) = 50000 \times (1.045)^{20}
$$

Calculating $$ (1.045)^{20} $$ (using a calculator or logarithmic approximation), we get approximately:

$$
(1.045)^{20} \approx 2.4116
$$

Thus, the account value after 20 years is:

$$
A(20) \approx 50000 \times 2.4116 \approx 120580
$$

So, Mary will have approximately \$120,580 in her account after 20 years.

---

**d. Situational Domain and Range**

- **Domain:**  
  Since $$ t $$ represents the number of years Mary has her money invested, and she plans to retire after 30 years, the domain is:

$$
  0 \leq t \leq 30 \quad \text{(years)}
  $$

- **Range:**  
  The smallest amount in the account is at $$ t = 0 $$, which is \$50,000, and the largest amount is at $$ t = 30 $$, calculated by:
  
  $$
  A(30) = 50000 \times (1.045)^{30}
  $$
  
  Therefore, the range for the account value is:

$$
  50000 \leq A(t) \leq 50000 \times (1.045)^{30} \quad \text{(dollars)}
  $$
**a. Variables** - **Independent Variable (IV):** $$ t $$ – Time in years, starting from when Mary invested \$50,000. Units: Years - **Dependent Variable (DV):** $$ A(t) $$ – The amount of money in the retirement account at time $$ t $$. Units: Dollars --- **b. Function Equation** $$ A(t) = 50000 \times (1.045)^t $$ --- **c. Amount After 20 Years** Mary will have approximately \$120,580 in her account after 20 years. --- **d. Domain and Range** - **Domain:** $$ 0 \leq t \leq 30 $$ years - **Range:** \$50,000 ≤ $$ A(t) $$ ≤ \$120,580

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