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?

Question

Play A: y = 15x + 76

Play B: y = –x2 + 36x – 4

Answer 1

To find the day(s) when attendance was the same at both plays, we need to set the attendance equations equal to each other and solve for x:

15x + 76 = -x^2 + 36x - 4

Rearrange the equation to set it equal to 0:

x^2 - 21x + 80 = 0

Factor the quadratic equation:

(x - 16)(x - 5) = 0

Set each factor equal to 0 and solve for x:

x - 16 = 0
x = 16

x - 5 = 0
x = 5

So, attendance was the same at both plays on day 5 and day 16. To find the attendance on those days, substitute the value of x back into either of the attendance equations:

For x = 5:
Play A: y = 15(5) + 76 = 151
Play B: y = -5^2 + 36(5) - 4 = 51

For x = 16:
Play A: y = 15(16) + 76 = 316
Play B: y = -16^2 + 36(16) - 4 = 316

Therefore, the attendance was the same at 151 on day 5 and at 316 on day 16 for both plays.