Certainly! Let's solve this problem step by step.
First, let's identify the values in the problem and the given dataset. We need to find the average rate of change for the interval from [tex]\(x = 8\)[/tex] to [tex]\(x = 9\)[/tex].
Given the exponential nature of the function and values in the table, observe that each [tex]\(y\)[/tex]-value is obtained by multiplying the previous [tex]\(y\)[/tex]-value by 3. This can be expressed as [tex]\( y = 3^x \)[/tex].
Next, let's find the [tex]\(y\)[/tex]-values for [tex]\(x = 8\)[/tex] and [tex]\(x = 9\)[/tex]:
- For [tex]\(x = 8\)[/tex], [tex]\( y = 3^8 \)[/tex].
- For [tex]\(x = 9\)[/tex], [tex]\( y = 3^9 \)[/tex].
Using this relationship, we calculate:
- [tex]\( y(8) = 3^8 = 6561 \)[/tex]
- [tex]\( y(9) = 3^9 = 19683 \)[/tex]
Now, the average rate of change over the interval [tex]\([8, 9]\)[/tex] is found using the formula for the rate of change:
[tex]\[ \text{Average Rate of Change} = \frac{y(9) - y(8)}{9 - 8} \][/tex]
Substitute the values we calculated:
[tex]\[ \text{Average Rate of Change} = \frac{19683 - 6561}{9 - 8} \][/tex]
[tex]\[ \text{Average Rate of Change} = \frac{13122}{1} \][/tex]
[tex]\[ \text{Average Rate of Change} = 13122 \][/tex]
From the options provided:
A. 6561
B. 13,122
C. 3
D. 19,683
The correct answer is [tex]\( 13,122 \)[/tex], which corresponds to option B.
