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=7x+84
Play B: y=-x^2+40x-6
A. The attendance was the same on day 30. The attendance was 294 at both place that day.
B.The attendance was the same on day 3. The attendance was 105 at both place that day.
C. The attendance was never the same at both plays
D. The attendance was the same on days 3 and 30. The attendance at both plays on those days was 105 and 294 respectively. ***
To find the day(s) on which the attendance was the same at both plays, we need to solve the equation:
7x + 84 = -x^2 + 40x - 6
Rearranging the equation, we get:
x^2 + 33x - 90 = 0
Now, we can factorize the equation:
(x - 3)(x + 30) = 0
Therefore, x = 3 or x = -30.
Since the number of days cannot be negative, we discard x = -30.
Thus, the attendance was the same on day 3.
To find the attendance on this day, let's substitute x = 3 into one of the equations:
y = 7x + 84
y = 7(3) + 84
y = 21 + 84
y = 105
Therefore, the attendance at both plays on day 3 was 105.
The correct answer is option B.
To find the day(s) when the attendance was the same at both plays, we need to set the attendance equations equal to each other and solve for x.
The equations are:
Play A: y = 7x + 84
Play B: y = -x^2 + 40x - 6
Setting them equal to each other gives:
7x + 84 = -x^2 + 40x - 6
Rearranging the equation to standard form:
x^2 - 33x + 90 = 0
Now we need to solve this quadratic equation. We can factor it:
(x - 3)(x - 30) = 0
Set each factor equal to zero:
x - 3 = 0 -> x = 3
x - 30 = 0 -> x = 30
So, the attendance was the same on days 3 and 30.
Now, let's find the attendance on those days. Substitute the values of x in the attendance equations to find y.
For day 3 (x = 3):
Play A: y = 7(3) + 84 = 21 + 84 = 105
Play B: y = -(3^2) + 40(3) - 6 = -9 + 120 - 6 = 105
For day 30 (x = 30):
Play A: y = 7(30) + 84 = 210 + 84 = 294
Play B: y = -(30^2) + 40(30) - 6 = -900 + 1200 - 6 = 294
Therefore, the attendance was the same on days 3 and 30, and the attendance at both plays on those days was 105 and 294, respectively.
So the answer is D. The attendance was the same on days 3 and 30. The attendance at both plays on those days was 105 and 294, respectively.
Days with the same attendance:
7x + 84 = -x^2 + 40x - 6
x^2 -33x + 90 = 0
(x-30)(x-3) = 0
The attendance was the same on days 3 and 30.
sub into the easier of the two equations to find the actual attendance.