Question
The values in the table represent the exponential growth of the population of a rural region, p, in thousands, t years since 2015.
t p
0 150
1 180
2 216
3 259
4 311
5 373
6 448
7 537
8 645
To the nearest whole number, what is the average rate of change over the interval [2, 8]
? What does the average rate of change indicate?
(1 point)
Responses
The average rate of change is −72
. On average, the revenue of the company decreases by 72,000 for every increase of 1 year over the interval [2, 8]
.
The average rate of change is negative 72 . On average, the revenue of the company decreases by 72,000 for every increase of 1 year over the interval left bracket 2 comma 8 right bracket .
The average rate of change is −72
. On average, the revenue of the company decreases by 72 for every increase of 1 year over the interval [2, 8]
.
The average rate of change is negative 72 . On average, the revenue of the company decreases by 72 for every increase of 1 year over the interval left bracket 2 comma 8 right bracket .
The average rate of change is 72. On average, the population of the region increases by 72 people for every increase of 1 year over the interval [2, 8]
.
The average rate of change is 72. On average, the population of the region increases by 72 people for every increase of 1 year over the interval left bracket 2 comma 8 right bracket .
The average rate of change is 72. On average, the population of the region increases by 72,000 people for every increase of 1 year over the interval [2, 8]
.
t p
0 150
1 180
2 216
3 259
4 311
5 373
6 448
7 537
8 645
To the nearest whole number, what is the average rate of change over the interval [2, 8]
? What does the average rate of change indicate?
(1 point)
Responses
The average rate of change is −72
. On average, the revenue of the company decreases by 72,000 for every increase of 1 year over the interval [2, 8]
.
The average rate of change is negative 72 . On average, the revenue of the company decreases by 72,000 for every increase of 1 year over the interval left bracket 2 comma 8 right bracket .
The average rate of change is −72
. On average, the revenue of the company decreases by 72 for every increase of 1 year over the interval [2, 8]
.
The average rate of change is negative 72 . On average, the revenue of the company decreases by 72 for every increase of 1 year over the interval left bracket 2 comma 8 right bracket .
The average rate of change is 72. On average, the population of the region increases by 72 people for every increase of 1 year over the interval [2, 8]
.
The average rate of change is 72. On average, the population of the region increases by 72 people for every increase of 1 year over the interval left bracket 2 comma 8 right bracket .
The average rate of change is 72. On average, the population of the region increases by 72,000 people for every increase of 1 year over the interval [2, 8]
.
Answers
Answered by
GPT-4o mini
To calculate the average rate of change of the population \( p \) over the interval \([2, 8]\), we use the formula:
\[
\text{Average Rate of Change} = \frac{p(8) - p(2)}{8 - 2}
\]
From the given data, we have:
- \( p(2) = 216 \) (for \( t = 2 \))
- \( p(8) = 645 \) (for \( t = 8 \))
Now, substituting these values into the formula:
\[
\text{Average Rate of Change} = \frac{645 - 216}{8 - 2} = \frac{429}{6} \approx 71.5
\]
Rounding to the nearest whole number gives us:
\[
\text{Average Rate of Change} \approx 72
\]
### Interpretation:
The average rate of change of 72 indicates that, on average, the population of the region increases by 72 people for every 1 year increase in time over the interval from 2017 (when \( t = 2 \)) to 2023 (when \( t = 8 \)). Therefore, based on the given options, the correct response is:
**The average rate of change is 72. On average, the population of the region increases by 72 people for every increase of 1 year over the interval [2, 8].**
\[
\text{Average Rate of Change} = \frac{p(8) - p(2)}{8 - 2}
\]
From the given data, we have:
- \( p(2) = 216 \) (for \( t = 2 \))
- \( p(8) = 645 \) (for \( t = 8 \))
Now, substituting these values into the formula:
\[
\text{Average Rate of Change} = \frac{645 - 216}{8 - 2} = \frac{429}{6} \approx 71.5
\]
Rounding to the nearest whole number gives us:
\[
\text{Average Rate of Change} \approx 72
\]
### Interpretation:
The average rate of change of 72 indicates that, on average, the population of the region increases by 72 people for every 1 year increase in time over the interval from 2017 (when \( t = 2 \)) to 2023 (when \( t = 8 \)). Therefore, based on the given options, the correct response is:
**The average rate of change is 72. On average, the population of the region increases by 72 people for every increase of 1 year over the interval [2, 8].**