Since opening night, attendance at Play A has increased steadily, while Play B first rose 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 and what was the attendance?
Play A: y=15x+76
Play B: y=-x^2 +36x-4
Option A) the attendance was the same on day 5. The attendance was 151 at both plays on that day.
Option B) the attendance was the same on day 16. The attendance was 316 at both plays on that day.
Option C) the attendance was the same on day 5 and 16. The attendance at both plays on that day was 151 and 316 respectively.
Option D) the attendance was never the same at both plays.
We need to find the values of x for which y is the same for both plays A and B. In other words, we need to solve the equation:
15x+76 = -x^2 +36x-4
Simplifying and rearranging:
x^2 -21x + 80 = 0
Factoring:
(x-5)(x-16) = 0
So the attendance is the same at both plays on day 5 and day 16. To find the attendance on those days, we can substitute them into either equation:
On day 5:
y = 15(5) + 76 = 151
y = -5^2 + 36(5) - 4 = 151
On day 16:
y = 15(16) + 76 = 316
y = -16^2 + 36(16) - 4 = 316
Therefore, the answer is option C) the attendance was the same on day 5 and 16. The attendance at both plays on that day was 151 and 316 respectively.