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
and
The exponential decay function to model the situation is .
The average rate of change over is . The averagervate of change over is . The rate
of change as increases.
(Round to three decimal places as needed.)

Ask by Bernard Mccoy. in the United States

Jan 24,2025

Question

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
and
The exponential decay function to model the situation is .
The average rate of change over is . The averagervate of change over is . The rate
of change as increases.
(Round to three decimal places as needed.)

Ask by Bernard Mccoy. in the United States

Jan 24,2025

UpStudy AI · Accepted Answer

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

### Exponential Decay Function
Given:
- **Initial value (f(0))**: 59
- **Decay factor**: 0.9

The general form of an exponential decay function is:
$$ f(x) = f(0) \cdot (\text{decay factor})^x $$

Substituting the given values:
$$ f(x) = 59 \cdot (0.9)^x $$

### Calculating Average Rates of Change

The **average rate of change** over an interval $$[a, b]$$ is calculated using the formula:
$$ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} $$

#### 1. Interval $$1 \leq x \leq 3$$
- **Calculate $$f(1)$$**:
  $$ f(1) = 59 \cdot (0.9)^1 = 59 \cdot 0.9 = 53.1 $$
  
- **Calculate $$f(3)$$**:
  $$ f(3) = 59 \cdot (0.9)^3 = 59 \cdot 0.729 = 43.011 $$
  
- **Average Rate of Change**:
  $$
  \frac{f(3) - f(1)}{3 - 1} = \frac{43.011 - 53.1}{2} = \frac{-10.089}{2} = -5.045
  $$
  
  **Rounded to three decimal places**: $$-5.045$$

#### 2. Interval $$4 \leq x \leq 6$$
- **Calculate $$f(4)$$**:
  $$ f(4) = 59 \cdot (0.9)^4 = 59 \cdot 0.6561 = 38.7099 $$
  
- **Calculate $$f(6)$$**:
  $$ f(6) = 59 \cdot (0.9)^6 = 59 \cdot 0.531441 = 31.355 $$
  
- **Average Rate of Change**:
  $$
  \frac{f(6) - f(4)}{6 - 4} = \frac{31.355 - 38.7099}{2} = \frac{-7.3549}{2} = -3.72745
  $$
  
  **Rounded to three decimal places**: $$-3.727$$

### Comparison of Rates of Change
- **First Interval ($$1 \leq x \leq 3$$)**: $$-5.045$$
- **Second Interval ($$4 \leq x \leq 6$$)**: $$-3.727$$

The **rate of change** becomes **less negative** as $$x$$ increases, indicating that the function is **decreasing at a slower rate** over the second interval compared to the first.

### Summary
- **Exponential Decay Function**: $$ f(x) = 59(0.9)^x $$
- **Average Rate of Change over $$1 \leq x \leq 3$$**: $$-5.045$$
- **Average Rate of Change over $$4 \leq x \leq 6$$**: $$-3.727$$
- **Rate of Change Behavior**: The rate of change **increases** as $$x$$ increases.
The exponential decay function is $$ f(x) = 59(0.9)^x $$. The average rate of change over $$1 \leq x \leq 3$$ is $$-5.045$$, and over $$4 \leq x \leq 6$$ is $$-3.727$$. As $$x$$ increases, the rate of change becomes less negative, meaning the function decreases at a slower rate.

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