Question
Since the season premiere of a TV series, the number of viewers of Show N first increased and then decreased. The equation modeling the daily viewers y of the show is given, where x is the number of days since the premiere.
y = −x^2+ 49x + 11
Which statement BEST describes the situation?
On day 7, there were 305 viewers who watched the show.
On day 7, there were 403 viewers who watched the show.
On day 305, there were 7 viewers who watched the show.
On day 403, there were 7 viewers who watched the show.
y = −x^2+ 49x + 11
Which statement BEST describes the situation?
On day 7, there were 305 viewers who watched the show.
On day 7, there were 403 viewers who watched the show.
On day 305, there were 7 viewers who watched the show.
On day 403, there were 7 viewers who watched the show.
Answers
GPT-4o mini
To analyze the situation described by the equation \( y = -x^2 + 49x + 11 \), we need to evaluate this quadratic equation at the specified days.
### Step 1: Calculate the number of viewers on day 7.
Plug \( x = 7 \) into the equation:
\[
y = -7^2 + 49(7) + 11
\]
\[
y = -49 + 343 + 11
\]
\[
y = 305
\]
### Step 2: Check day 7 options:
The first option states:
- **On day 7, there were 305 viewers who watched the show.** (This is **True** based on our calculation.)
The second option states:
- **On day 7, there were 403 viewers who watched the show.** (This is **False** based on our calculation.)
### Step 3: Calculate the number of viewers on day 305.
Plug \( x = 305 \) into the equation:
\[
y = -305^2 + 49(305) + 11
\]
Calculating \( 305^2 \) first:
\[
-305^2 = -93025
\]
Calculating \( 49 \times 305 \):
\[
49 \times 305 = 14945
\]
Now plug these values into the equation:
\[
y = -93025 + 14945 + 11
\]
\[
y = -93025 + 14956
\]
\[
y = -78069
\]
Since the number of viewers cannot be negative, there would be **0 viewers**.
### Step 4: Check day 305 option:
The third option states:
- **On day 305, there were 7 viewers who watched the show.** (This is **False** based on our calculation; the number is actually negative, not feasible.)
The fourth option states:
- **On day 403, there were 7 viewers who watched the show.**
As shown, we find \( y \) at \( x = 403 \),
\[
y = -403^2 + 49(403) + 11
\]
Calculating \( 403^2 \):
\[
-403^2 = -162409
\]
Calculating \( 49 \times 403 \):
\[
49 \times 403 = 19747
\]
Now plug these values into the equation:
\[
y = -162409 + 19758
\]
\[
y = -142651
\]
### Final Assessment: All options evaluated
- **On day 7, there were 305 viewers who watched the show.** (True)
- **On day 7, there were 403 viewers who watched the show.** (False)
- **On day 305, there were 7 viewers who watched the show.** (False)
- **On day 403, there were 7 viewers who watched the show.** (False)
Thus, the BEST statement that describes the situation is:
**On day 7, there were 305 viewers who watched the show.**
### Step 1: Calculate the number of viewers on day 7.
Plug \( x = 7 \) into the equation:
\[
y = -7^2 + 49(7) + 11
\]
\[
y = -49 + 343 + 11
\]
\[
y = 305
\]
### Step 2: Check day 7 options:
The first option states:
- **On day 7, there were 305 viewers who watched the show.** (This is **True** based on our calculation.)
The second option states:
- **On day 7, there were 403 viewers who watched the show.** (This is **False** based on our calculation.)
### Step 3: Calculate the number of viewers on day 305.
Plug \( x = 305 \) into the equation:
\[
y = -305^2 + 49(305) + 11
\]
Calculating \( 305^2 \) first:
\[
-305^2 = -93025
\]
Calculating \( 49 \times 305 \):
\[
49 \times 305 = 14945
\]
Now plug these values into the equation:
\[
y = -93025 + 14945 + 11
\]
\[
y = -93025 + 14956
\]
\[
y = -78069
\]
Since the number of viewers cannot be negative, there would be **0 viewers**.
### Step 4: Check day 305 option:
The third option states:
- **On day 305, there were 7 viewers who watched the show.** (This is **False** based on our calculation; the number is actually negative, not feasible.)
The fourth option states:
- **On day 403, there were 7 viewers who watched the show.**
As shown, we find \( y \) at \( x = 403 \),
\[
y = -403^2 + 49(403) + 11
\]
Calculating \( 403^2 \):
\[
-403^2 = -162409
\]
Calculating \( 49 \times 403 \):
\[
49 \times 403 = 19747
\]
Now plug these values into the equation:
\[
y = -162409 + 19758
\]
\[
y = -142651
\]
### Final Assessment: All options evaluated
- **On day 7, there were 305 viewers who watched the show.** (True)
- **On day 7, there were 403 viewers who watched the show.** (False)
- **On day 305, there were 7 viewers who watched the show.** (False)
- **On day 403, there were 7 viewers who watched the show.** (False)
Thus, the BEST statement that describes the situation is:
**On day 7, there were 305 viewers who watched the show.**