To compare the average rate of change from [tex]\( x = 0 \)[/tex] to [tex]\( x = 20 \)[/tex] with the given average rate of change from [tex]\( x = 0 \)[/tex] to [tex]\( x = 15 \)[/tex], let's start by determining the average rate of change over these intervals. Here is the detailed step-by-step solution:
1. Given Information:
- The average rate of change from [tex]\( x = 0 \)[/tex] to [tex]\( x = 15 \)[/tex] is approximately [tex]\(-4.667\)[/tex].
- The height of the slide at [tex]\( x = 20 \)[/tex] is 2 feet.
2. Determine the Average Rate of Change from [tex]\( x = 0 \)[/tex] to [tex]\( x = 20 \)[/tex]:
- To find the average rate of change over an interval [tex]\([x_1, x_2]\)[/tex], we use the formula:
[tex]\[
\text{Average Rate of Change} = \frac{\text{Change in Value}}{\text{Change in Time}}
\][/tex]
- For the interval [tex]\( x = 0 \)[/tex] to [tex]\( x = 20 \)[/tex]:
[tex]\[
\text{Average Rate of Change} = \frac{\text{Height at } x = 20 - \text{Height at } x = 0}{20 - 0} = \frac{2 - 0}{20} = \frac{2}{20} = 0.1
\][/tex]
3. Comparison of Average Rates of Change:
- The average rate of change from [tex]\( x = 0 \)[/tex] to [tex]\( x = 15 \)[/tex] is [tex]\(-4.667\)[/tex].
- The average rate of change from [tex]\( x = 0 \)[/tex] to [tex]\( x = 20 \)[/tex] is [tex]\(0.1\)[/tex].
- The difference between these two average rates of change is:
[tex]\[
\text{Difference} = -4.667 - 0.1 = -4.767
\][/tex]
4. Conclusions:
- The average rate of change from [tex]\( x = 0 \)[/tex] to [tex]\( x = 20 \)[/tex] is [tex]\(0.1\)[/tex].
- The average rate of change from [tex]\( x = 0 \)[/tex] to [tex]\( x = 20 \)[/tex] is decreasing slower than the average rate of change from [tex]\( x = 0 \)[/tex] to [tex]\( x = 15\)[/tex] by [tex]\(-4.767\)[/tex].
Summary:
- The average rate of change from [tex]\( x = 0 \)[/tex] to [tex]\( x = 20 \)[/tex] is [tex]\(0.1\)[/tex].
- The average rate of change from [tex]\( x = 0 \)[/tex] to [tex]\( x = 20 \)[/tex] is decreasing [tex]\(-4.767\)[/tex] slower than the average rate of change from [tex]\( x = 0 \)[/tex] to [tex]\( x = 15\)[/tex].
Ultimately:
- The average rate of change over the entire slide from [tex]\( x = 0 \)[/tex] to [tex]\( x = 20 \)[/tex] is [tex]\(0.1\)[/tex].
