Suppose that the dollar value $$ v(t) $$ of a certain car that is $$ t $$ years old is given by the following exponential function. $$ \begin{array}{l}v(t)=29,900(0.91)^{t} \ \text { Find the initial value of the car. } \ \text { Does the function represent growth or decay? } \ \text { O growth } \quad \text { O decay } \ \text { By what percent does the value of the car change each year? } \ \square \%\end{array} $$

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UpStudy AI · Accepted Answer

Let's address each part of the problem step by step:

### 1. **Initial Value of the Car**

The initial value of the car is determined by evaluating the value function $$ v(t) $$ at $$ t = 0 $$:

$$
v(0) = 29,900 \times (0.91)^0 = 29,900 \times 1 = \boxed{29,\!900\ \text{dollars}}
$$

So, **the initial value of the car is \$29,900**.

---

### 2. **Growth or Decay**

The function given is:

$$
v(t) = 29,900 \times (0.91)^t
$$

Here, the base of the exponential function is **0.91**, which is less than **1**.

- **If the base is greater than 1**, the function represents **growth**.
- **If the base is between 0 and 1**, the function represents **decay**.

Since **0.91 < 1**, the function represents **decay**.

**Answer:**  
☑ **Decay**  
☐ Growth

---

### 3. **Annual Percentage Change**

The base of the exponential function **0.91** indicates the factor by which the value changes each year. To find the percentage change:

$$
\text{Percentage Change} = (0.91 - 1) \times 100\% = -0.09 \times 100\% = -9\%
$$

The negative sign indicates a decrease.

**Answer:**  
$$\boxed{9\%}$$ (The value of the car **decreases by 9%** each year)

---

### **Summary of Answers**
1. **Initial Value:** \$29,900
2. **Function Represents:** Decay
3. **Annual Percentage Change:** 9% decrease
- **Initial Value:** \$29,900 - **Function Represents:** Decay - **Annual Percentage Change:** 9% decrease

Suppose that the dollar value of a certain car that is years old is given by the following exponential function.

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