On what days was the attendance the same at both plays? What was the attendance?
Play A:y=7x+84
Play B:y=-x^2+40x-6
To find the days on which the attendance was the same at both plays, we need to set the attendance functions equal to each other and solve for x.
So, we have:
7x + 84 = -x^2 + 40x - 6
Rearranging the equation to have all terms on one side:
x^2 + 33x - 90 = 0
Now, we can solve this quadratic equation by factoring or using the quadratic formula. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = 1, b = 33, and c = -90, so we can substitute these values into the formula:
x = (-33 ± √(33^2 - 4(1)(-90))) / (2(1))
Simplifying:
x = (-33 ± √(1089 + 360)) / 2
x = (-33 ± √1449) / 2
Now, we have two possible solutions for x:
x₁ = (-33 + √1449) / 2
x₂ = (-33 - √1449) / 2
Calculating each solution:
x₁ ≈ 1.607
x₂ ≈ -34.607
Since the days can't be negative, we can discard the negative value.
Therefore, the attendance will be the same at both plays on approximately day 1.607.
To find the attendance, we substitute this value of x into either of the attendance functions. Let's use Play A:
y = 7(1.607) + 84
y ≈ 95.25
So, the attendance is approximately 95.25 at both plays on the same day.