(x=1 is the day tickets go on sale)
Tickets= -0.2x^2+12x+11
1) Use the quadratic equation to determine the last day that tickets will be sold.
Note. Write your answer in terms of the number of days after ticket sales begin.
2) Will tickets peak or be at a low during the middle of the sale? How do you know?
For this problem I already solved for the peak value that will occur on day 30 by using the formula -b/(2a).
When x is 30 the maximum value of y is 191. right?
To find the last day the tickets could be on sale I need to factor to find the two values in which the value of x is 0. (or use the quadratic formula) I'm not sure I did this right because it doesn't look right. can someone help and verify if I am on the right track?
-0.2x^2+12x+11
When x = 30, y = 191, so yes, you are correct.
Solve this equation by completing the square,
-0.2x^2+12x+11
Divide both sides by -0.2
x^2 - 60x - 55 = 0
x^2 - 60x = 55
x^2 - 60x + 900 = 55 + 900
(x - 30)^2 = 955
+ - (x - 30) = (sqrt(955))
+ x - 30 = 30.9031
x = 60.9031
- x + 30 = 30.9031
- x = 0.9031
x = -0.9031
To find the last day that tickets will be sold, we need to determine the values of x when Tickets equals zero. This can be done by using the quadratic formula or factoring the quadratic expression.
Let's use the quadratic formula to solve for x:
The quadratic equation is: -0.2x^2 + 12x + 11 = 0
The quadratic formula is: x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = -0.2, b = 12, and c = 11.
Plugging in these values, we get:
x = (-12 ± √(12^2 - 4*(-0.2)*11)) / (2*(-0.2))
Simplifying further:
x = (-12 ± √(144 + 8.8)) / (-0.4)
x = (-12 ± √(152.8)) / (-0.4)
x = (-12 ± 12.36) / (-0.4)
To find the last day that tickets will be sold, we need the positive value:
x = (-12 + 12.36) / (-0.4)
x ≈ 0.9
Therefore, the last day that tickets will be sold is approximately 0.9 days after ticket sales begin.
Now let's move on to the second question.
The given quadratic equation for Tickets, -0.2x^2 + 12x + 11, represents a downward-opening parabola since the coefficient of x^2 is negative (-0.2 < 0). In such cases, the vertex of the parabola represents either the highest or lowest point of the graph.
The x-coordinate of the vertex can be found using the formula -b / (2a). In this case, a = -0.2 and b = 12.
x = -12 / (2 * -0.2)
x = -12 / -0.4
x = 30
To find the value of Tickets at x = 30:
Tickets = -0.2(30)^2 + 12(30) + 11
Tickets = -0.2(900) + 360 + 11
Tickets = -180 + 360 + 11
Tickets = 191
Therefore, when x = 30, the maximum value of Tickets is indeed 191.
Since the vertex occurs at x = 30 and the Tickets value reaches its maximum of 191 at that point, we can determine that the tickets will peak during the middle of the sale.