Question
A theater plans to increase their ticket prices by x dollars. The function t (x) = 350 - 10x represents the number of tickets the theater expects to sell and r(x) = (350 - 10x) (10-x) represents the theater's expected revenue from selling the tickets with the new price.
Abigail determined the average rate of change for both functions over the interval [11,15] is -10. She claims this means both the number of tickets sold and revenue from tickets are expected to decrease by 10 for each dollar increase in ticket prices for all values of x.
Which statement BEST explains Abigail's error?
Group of answer choices
Abigail incorrectly interpreted the meaning of the common average rate of change. While the average rate of change for revenue from tickets will be -10 for all intervals, the average rate of change for number of tickets sold will not be -10 for all intervals.
Abigail incorrectly interpreted the meaning of the common average rate of change. While the average rate of change for the number of tickets sold will be -10 for all intervals, the average rate of change for revenue from tickets will not be -10 for all intervals.
Abigail incorrectly calculated the average rate of change of the functions. The average rate of change of both functions over the interval [11,15] is 10. Both the number of tickets sold and revenue from tickets are expected to increase by 10 for each dollar increase in ticket price.
Abigail incorrectly calculated the average rate of change of the functions. The average rate of change of both functions over the interval [11,15] is -40. Both the number of tickets sold and revenue from tickets are expected to decrease by 40 for each dollar increase in ticket prices.
Abigail determined the average rate of change for both functions over the interval [11,15] is -10. She claims this means both the number of tickets sold and revenue from tickets are expected to decrease by 10 for each dollar increase in ticket prices for all values of x.
Which statement BEST explains Abigail's error?
Group of answer choices
Abigail incorrectly interpreted the meaning of the common average rate of change. While the average rate of change for revenue from tickets will be -10 for all intervals, the average rate of change for number of tickets sold will not be -10 for all intervals.
Abigail incorrectly interpreted the meaning of the common average rate of change. While the average rate of change for the number of tickets sold will be -10 for all intervals, the average rate of change for revenue from tickets will not be -10 for all intervals.
Abigail incorrectly calculated the average rate of change of the functions. The average rate of change of both functions over the interval [11,15] is 10. Both the number of tickets sold and revenue from tickets are expected to increase by 10 for each dollar increase in ticket price.
Abigail incorrectly calculated the average rate of change of the functions. The average rate of change of both functions over the interval [11,15] is -40. Both the number of tickets sold and revenue from tickets are expected to decrease by 40 for each dollar increase in ticket prices.
Answers
Answered by
GPT-4o mini
To accurately assess Abigail's claim regarding the average rate of change for the functions, we first need to examine both functions, \(t(x)\) and \(r(x)\):
1. **Number of tickets sold:**
\[
t(x) = 350 - 10x
\]
2. **Revenue from tickets:**
\[
r(x) = (350 - 10x)(10 - x)
\]
Next, we find the average rate of change of both functions over the interval \([11, 15]\).
**Step 1: Calculate \(t(11)\) and \(t(15)\)**
\[
t(11) = 350 - 10(11) = 350 - 110 = 240
\]
\[
t(15) = 350 - 10(15) = 350 - 150 = 200
\]
\[
\text{Average rate of change of } t \text{ on } [11, 15] = \frac{t(15) - t(11)}{15 - 11} = \frac{200 - 240}{4} = \frac{-40}{4} = -10
\]
**Step 2: Calculate \(r(11)\) and \(r(15)\)**
\[
r(11) = (350 - 10(11))(10 - 11) = (350 - 110)(-1) = 240 \cdot -1 = -240
\]
\[
r(15) = (350 - 10(15))(10 - 15) = (350 - 150)(-5) = 200 \cdot -5 = -1000
\]
\[
\text{Average rate of change of } r \text{ on } [11, 15] = \frac{r(15) - r(11)}{15 - 11} = \frac{-1000 - (-240)}{4} = \frac{-1000 + 240}{4} = \frac{-760}{4} = -190
\]
Both rates of change are not equal, and importantly, they vary significantly.
Now, Abigail's claim that both the number of tickets sold and the revenue will change by -10 for each dollar increase in ticket prices is incorrect for the revenue function, as we've now established a much larger average rate of change.
**Analysis of the answer choices:**
- The first choice states: "Abigail incorrectly interpreted the meaning of the common average rate of change. While the average rate of change for revenue from tickets will be -10 for all intervals, the average rate of change for the number of tickets sold will not be -10 for all intervals." This is incorrect since both average rates of change we calculated are -10 for tickets, but they do **not** match with revenue.
- The second choice states: "Abigail incorrectly interpreted the meaning of the common average rate of change. While the average rate of change for the number of tickets sold will be -10 for all intervals, the average rate of change for revenue from tickets will not be -10 for all intervals." This choice defines the distinction correctly between the two functions but is incorrect because we found different values.
- The third choice is about an incorrect calculation of -10 for both, which is misleading since the calculation for revenue did not yield -10.
- The fourth choice indicates that she has calculated the average rate incorrectly with a rate of -40 which again, based on our calculations, is not true.
Thus, it's clear that Abigail's assumptions about the constancy of the average rate of change for both functions across all intervals is what needs addressing. Her error stems from the incorrect generalization based on one segment of her calculations.
The BEST explanation of the error is:
**Abigail incorrectly interpreted the meaning of the common average rate of change. While the average rate of change for revenue from tickets will not be consistent and is indeed negative (-190), the average rate of change for the number of tickets sold remains consistent at -10 for the specified interval.** Hence, neither of the answer choices is completely accurate based on the calculated data, but the reasoning should be focused on the inconsistency in revenue change versus ticket numbers.
However, the closest aligned option regarding confusion between the two functions (specifically that of revenue) would be the:
**Abigail incorrectly interpreted the meaning of the common average rate of change. While the average rate of change for revenue will differ, that for tickets sold will be constant over the interval.**
1. **Number of tickets sold:**
\[
t(x) = 350 - 10x
\]
2. **Revenue from tickets:**
\[
r(x) = (350 - 10x)(10 - x)
\]
Next, we find the average rate of change of both functions over the interval \([11, 15]\).
**Step 1: Calculate \(t(11)\) and \(t(15)\)**
\[
t(11) = 350 - 10(11) = 350 - 110 = 240
\]
\[
t(15) = 350 - 10(15) = 350 - 150 = 200
\]
\[
\text{Average rate of change of } t \text{ on } [11, 15] = \frac{t(15) - t(11)}{15 - 11} = \frac{200 - 240}{4} = \frac{-40}{4} = -10
\]
**Step 2: Calculate \(r(11)\) and \(r(15)\)**
\[
r(11) = (350 - 10(11))(10 - 11) = (350 - 110)(-1) = 240 \cdot -1 = -240
\]
\[
r(15) = (350 - 10(15))(10 - 15) = (350 - 150)(-5) = 200 \cdot -5 = -1000
\]
\[
\text{Average rate of change of } r \text{ on } [11, 15] = \frac{r(15) - r(11)}{15 - 11} = \frac{-1000 - (-240)}{4} = \frac{-1000 + 240}{4} = \frac{-760}{4} = -190
\]
Both rates of change are not equal, and importantly, they vary significantly.
Now, Abigail's claim that both the number of tickets sold and the revenue will change by -10 for each dollar increase in ticket prices is incorrect for the revenue function, as we've now established a much larger average rate of change.
**Analysis of the answer choices:**
- The first choice states: "Abigail incorrectly interpreted the meaning of the common average rate of change. While the average rate of change for revenue from tickets will be -10 for all intervals, the average rate of change for the number of tickets sold will not be -10 for all intervals." This is incorrect since both average rates of change we calculated are -10 for tickets, but they do **not** match with revenue.
- The second choice states: "Abigail incorrectly interpreted the meaning of the common average rate of change. While the average rate of change for the number of tickets sold will be -10 for all intervals, the average rate of change for revenue from tickets will not be -10 for all intervals." This choice defines the distinction correctly between the two functions but is incorrect because we found different values.
- The third choice is about an incorrect calculation of -10 for both, which is misleading since the calculation for revenue did not yield -10.
- The fourth choice indicates that she has calculated the average rate incorrectly with a rate of -40 which again, based on our calculations, is not true.
Thus, it's clear that Abigail's assumptions about the constancy of the average rate of change for both functions across all intervals is what needs addressing. Her error stems from the incorrect generalization based on one segment of her calculations.
The BEST explanation of the error is:
**Abigail incorrectly interpreted the meaning of the common average rate of change. While the average rate of change for revenue from tickets will not be consistent and is indeed negative (-190), the average rate of change for the number of tickets sold remains consistent at -10 for the specified interval.** Hence, neither of the answer choices is completely accurate based on the calculated data, but the reasoning should be focused on the inconsistency in revenue change versus ticket numbers.
However, the closest aligned option regarding confusion between the two functions (specifically that of revenue) would be the:
**Abigail incorrectly interpreted the meaning of the common average rate of change. While the average rate of change for revenue will differ, that for tickets sold will be constant over the interval.**