Write an exponential decay function to model the situation. Compare the average rates of change over the given intervals. initial value: 59 decay factor: 0.9 $$ 1 \leq x \leq 3 $$ and $$ 4 \leq x \leq 6 $$ The exponential decay function to model the situation is $$ f(x)=\square $$.

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To model the given situation using an exponential decay function and compare the average rates of change over the specified intervals, follow the steps below:

### **1. Exponential Decay Function**

The general form of an exponential decay function is:

$$ f(x) = A \cdot (D)^x $$

Where:
- $$ A $$ is the initial value.
- $$ D $$ is the decay factor.
- $$ x $$ is the independent variable (e.g., time).

Given:
- **Initial Value ($$ A $$)**: 59
- **Decay Factor ($$ D $$)**: 0.9

Substituting the given values into the general form:

$$ f(x) = 59 \cdot (0.9)^x $$

**Thus, the exponential decay function is:**

$$ f(x) = 59(0.9)^x $$

### **2. Average Rate of Change**

The **average rate of change** of a function $$ f(x) $$ over an interval $$[a, b]$$ is calculated using the formula:

$$ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} $$

We'll calculate this for the two intervals: $$ 1 \leq x \leq 3 $$ and $$ 4 \leq x \leq 6 $$.

#### **a. Interval $$ 1 \leq x \leq 3 $$**

1. **Calculate $$ f(1) $$:**
   $$ f(1) = 59 \cdot (0.9)^1 = 59 \cdot 0.9 = 53.1 $$

2. **Calculate $$ f(3) $$:**
   $$ f(3) = 59 \cdot (0.9)^3 = 59 \cdot 0.729 = 43.011 $$

3. **Average Rate of Change:**
   $$ \frac{f(3) - f(1)}{3 - 1} = \frac{43.011 - 53.1}{2} = \frac{-10.089}{2} = -5.0445 $$

**Rounded to two decimal places:** 
   $$ \approx -5.04 $$

#### **b. Interval $$ 4 \leq x \leq 6 $$**

1. **Calculate $$ f(4) $$:**
   $$ f(4) = 59 \cdot (0.9)^4 = 59 \cdot 0.6561 = 38.7099 $$

2. **Calculate $$ f(6) $$:**
   $$ f(6) = 59 \cdot (0.9)^6 = 59 \cdot 0.531441 = 31.353 $$

3. **Average Rate of Change:**
   $$ \frac{f(6) - f(4)}{6 - 4} = \frac{31.353 - 38.7099}{2} = \frac{-7.3569}{2} = -3.67845 $$

**Rounded to two decimal places:** 
   $$ \approx -3.68 $$

### **3. Summary**

- **Exponential Decay Function:** 
  $$ f(x) = 59(0.9)^x $$

- **Average Rates of Change:**
  - **For $$ 1 \leq x \leq 3 $$:** 
    $$ \approx -5.04 $$
  - **For $$ 4 \leq x \leq 6 $$:** 
    $$ \approx -3.68 $$

This comparison shows that the average rate of decay is decreasing (i.e., becoming less negative) as $$ x $$ increases, which is consistent with the nature of exponential decay.
The exponential decay function is $$ f(x) = 59(0.9)^x $$. Over the intervals $$ 1 \leq x \leq 3 $$, the average rate of change is approximately -5.04, and over $$ 4 \leq x \leq 6 $$, it is approximately -3.68.

Write an exponential decay function to model the situation. Compare the average rates of change over the given intervals. initial value: 59 decay factor: 0.9 \( 1 \leq x \leq 3 \) and \( 4 \leq x \leq 6 \) The exponential decay function to model the situation is \( f(x)=\square \).

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