I am supposed to find the ticket sales of a concert using this formula:
N(x)= -0.3x^2 + 6x + 15
X represtents the number of days since the concert was first announced. I need to find out when daily ticket sales will peak and how many tickets will be sold on that day. How do i work out this problem I don't know the first place to start! please help.
If you are not in Calculus, then I would graph the equation to find x for the maximum sales.
So I should find the vertex?
I assume no calculus, therefore we must look at the shape of this curve.
it is of form ax^2 + bx + c, a parabola
y = -.3 x^2 + 6 x + 15
we need to find the vertex
complete the square
y/-.3 = x^2 +6/-.3 x +15/-.3
x^2 - 20x -50 = -3.33 y
x^2 -20 x = -3.33 y +50
x^2 -20 x +100 = -3.33y +150
(x-10)^2) = -3.33 ( y - 45)
vertex at (10,45)
10 days, 45 tickets
To find out when the daily ticket sales will reach their peak and how many tickets will be sold on that day, you will need to determine the maximum value of the function N(x) = -0.3x^2 + 6x + 15.
The function represents a quadratic equation in terms of x. The peak of the graph occurs at the vertex of the quadratic equation.
To find the x-coordinate of the vertex, you can use the formula:
x = -b / (2a)
In the equation N(x) = -0.3x^2 + 6x + 15, the coefficient of x^2 is -0.3 and the coefficient of x is 6. Plugging these values into the formula, you get:
x = -6 / (2*(-0.3))
x = -6 / (-0.6)
x = 10
The x-coordinate of the vertex is 10, which corresponds to the number of days since the concert was first announced.
To find the y-coordinate of the vertex, substitute the x-coordinate back into the function:
N(x) = -0.3x^2 + 6x + 15
N(10) = -0.3(10)^2 + 6(10) + 15
N(10) = -0.3(100) + 60 + 15
N(10) = -30 + 60 + 15
N(10) = 45
The y-coordinate of the vertex is 45, which represents the number of tickets sold on the day when daily ticket sales will peak.
Therefore, the daily ticket sales will peak on day 10, and a total of 45 tickets will be sold on that day.