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.
a. find the average rate of change of the function over the intervals $6 < x < 8$ and $9 < x < 11$.
b. interpret the average rates of change
c. what does the difference in the average rates of change mean in terms of the situation?

a. the average rate of change of $c(x)$ over $6 < x < 8$ is $24.36$
(simplify your answer. type an integer or a decimal.)
the average rate of change of $c(x)$ over $9 < x < 11$ is $34.8$
(simplify your answer. type an integer or a decimal.)
b. between 6 and 8 inches, the calories in a tortilla $square$ by about $square$ calories per inch, and between 9 and 11 inches, the calories in a tortilla $square$ by about $square$ calories per inch

Explanation:

Part b

The average rate of change of a function over an interval represents how much the function's output (in this case, calories) changes per unit change in the input (diameter in inches). A positive average rate of change means the function is increasing over that interval.

For the interval \( 6 < x < 8 \), the average rate of change is 24.36, which is positive, so the calories in the tortilla increase. For the interval \( 9 < x < 11 \), the average rate of change is 34.8, also positive, so the calories increase here too. We use the values of the average rates of change (24.36 and 34.8) to fill in the blanks for the number of calories per inch.

The difference in the average rates of change (\( 34.8 - 24.36 = 10.44 \)) shows that the rate at which calories increase with respect to the diameter of the tortilla is higher when the diameter is between 9 and 11 inches compared to when it is between 6 and 8 inches. This means that as the tortilla's diameter gets larger (moving from the smaller interval to the larger interval), each additional inch in diameter results in a greater increase in the number of calories. In other words, the calorie - diameter relationship is such that larger - diameter tortillas (in the 9 - 11 inch range) have a steeper increase in calories per inch of diameter than smaller - diameter tortillas (in the 6 - 8 inch range).

Answer:

Between 6 and 8 inches, the calories in a tortilla \(\boldsymbol{\text{increase}}\) by about \(\boldsymbol{24.36}\) calories per inch, and between 9 and 11 inches, the calories in a tortilla \(\boldsymbol{\text{increase}}\) by about \(\boldsymbol{34.8}\) calories per inch.

Part c