Question
A burrito company uses the function C(x)=1.74x^2 to calculate the number of calories in a tortilla with a diameter of x inches.
Find the average rate of change of the function over the intervals 6<x<8 and 9<x<11, interpret the average rates of change, and what does the difference in the average rates of change mean in terms of the situation?
Find the average rate of change of the function over the intervals 6<x<8 and 9<x<11, interpret the average rates of change, and what does the difference in the average rates of change mean in terms of the situation?
Answers
GPT-4o mini
To find the average rate of change of the function \(C(x) = 1.74x^2\) over the specified intervals, we can use the formula for the average rate of change:
\[
\text{Average Rate of Change} = \frac{C(b) - C(a)}{b - a}
\]
where \(a\) and \(b\) are the endpoints of the interval.
### Interval \(6 < x < 8\)
1. Calculate \(C(6)\):
\[
C(6) = 1.74(6^2) = 1.74(36) = 62.64
\]
2. Calculate \(C(8)\):
\[
C(8) = 1.74(8^2) = 1.74(64) = 111.36
\]
3. Calculate the average rate of change from \(x = 6\) to \(x = 8\):
\[
\text{Average Rate of Change} = \frac{C(8) - C(6)}{8 - 6} = \frac{111.36 - 62.64}{2} = \frac{48.72}{2} = 24.36
\]
### Interval \(9 < x < 11\)
1. Calculate \(C(9)\):
\[
C(9) = 1.74(9^2) = 1.74(81) = 140.94
\]
2. Calculate \(C(11)\):
\[
C(11) = 1.74(11^2) = 1.74(121) = 210.54
\]
3. Calculate the average rate of change from \(x = 9\) to \(x = 11\):
\[
\text{Average Rate of Change} = \frac{C(11) - C(9)}{11 - 9} = \frac{210.54 - 140.94}{2} = \frac{69.60}{2} = 34.80
\]
### Interpretation and Difference in Average Rates of Change
1. **Average Rate of Change over \(6 < x < 8\)**: \(24.36\) calories per inch. This means that, on average, for each additional inch in diameter of the tortilla, the calorie count increases by \(24.36\) calories.
2. **Average Rate of Change over \(9 < x < 11\)**: \(34.80\) calories per inch. This indicates that for each additional inch in diameter of the tortilla in this interval, the calorie count increases by \(34.80\) calories.
### Difference in Average Rates of Change
The difference in the average rates of change indicates that as the diameter of the tortilla increases (specifically moving from the interval \(6 < x < 8\) to \(9 < x < 11\)), the increase in calories per additional inch of diameter also increases. This suggests that larger tortillas are significantly more calorie-dense than smaller ones, reflecting a non-linear growth in calories as diameter increases, which is consistent with the fact that the function \(C(x) = 1.74x^2\) is a quadratic function. This means that as the size of the tortilla increases, the rate of increase in calories accelerates.
\[
\text{Average Rate of Change} = \frac{C(b) - C(a)}{b - a}
\]
where \(a\) and \(b\) are the endpoints of the interval.
### Interval \(6 < x < 8\)
1. Calculate \(C(6)\):
\[
C(6) = 1.74(6^2) = 1.74(36) = 62.64
\]
2. Calculate \(C(8)\):
\[
C(8) = 1.74(8^2) = 1.74(64) = 111.36
\]
3. Calculate the average rate of change from \(x = 6\) to \(x = 8\):
\[
\text{Average Rate of Change} = \frac{C(8) - C(6)}{8 - 6} = \frac{111.36 - 62.64}{2} = \frac{48.72}{2} = 24.36
\]
### Interval \(9 < x < 11\)
1. Calculate \(C(9)\):
\[
C(9) = 1.74(9^2) = 1.74(81) = 140.94
\]
2. Calculate \(C(11)\):
\[
C(11) = 1.74(11^2) = 1.74(121) = 210.54
\]
3. Calculate the average rate of change from \(x = 9\) to \(x = 11\):
\[
\text{Average Rate of Change} = \frac{C(11) - C(9)}{11 - 9} = \frac{210.54 - 140.94}{2} = \frac{69.60}{2} = 34.80
\]
### Interpretation and Difference in Average Rates of Change
1. **Average Rate of Change over \(6 < x < 8\)**: \(24.36\) calories per inch. This means that, on average, for each additional inch in diameter of the tortilla, the calorie count increases by \(24.36\) calories.
2. **Average Rate of Change over \(9 < x < 11\)**: \(34.80\) calories per inch. This indicates that for each additional inch in diameter of the tortilla in this interval, the calorie count increases by \(34.80\) calories.
### Difference in Average Rates of Change
The difference in the average rates of change indicates that as the diameter of the tortilla increases (specifically moving from the interval \(6 < x < 8\) to \(9 < x < 11\)), the increase in calories per additional inch of diameter also increases. This suggests that larger tortillas are significantly more calorie-dense than smaller ones, reflecting a non-linear growth in calories as diameter increases, which is consistent with the fact that the function \(C(x) = 1.74x^2\) is a quadratic function. This means that as the size of the tortilla increases, the rate of increase in calories accelerates.