Since opening night, attendance at Play A has increased steadily, while attendance at Play B first rose and then fell. Equations modeling the daily attendance y at each play are shown below, where x is the number of days since opening night. On what day(s) was the attendance the same at both plays? What was the attendance?
Play A: y = 15x + 76
Play B: y = –x^2 + 36x – 4
answers
1 The attendance was the same on day 5. The attendance was 151 at both plays on that day.
2 The attendance was the same on day 16. The attendance was 316 at both plays on that day.
3 The attendance was the same on days 5 and 16. The attendance at both plays on those days was 151 and 316 respectively.
4 The attendance was never the same at both plays.
To find the day(s) when the attendance was the same at both plays, we need to set the equations equal to each other and solve for x.
Setting the equations equal to each other:
15x + 76 = -x^2 + 36x - 4
Rearranging the equation:
x^2 + 21x - 80 = 0
Factoring the equation:
(x + 16)(x - 5) = 0
Solving for x:
x + 16 = 0 or x - 5 = 0
x = -16 or x = 5
Since the number of days cannot be negative, we conclude that the attendance was the same on day 5.
Substituting x = 5 into either equation, we can find the attendance:
For Play A:
y = 15x + 76
y = 15(5) + 76
y = 75 + 76
y = 151
For Play B:
y = -x^2 + 36x - 4
y = -(5)^2 + 36(5) - 4
y = -25 + 180 - 4
y = 151
Therefore, the attendance was the same on day 5, and the attendance at both plays on that day was 151.
The correct answer is option 1: The attendance was the same on day 5. The attendance was 151 at both plays on that day.