The number of tickets sold each day for an upcoming performance is given by N(x)=-0.3x^2+9x+15, where x is the number of days since the concert was first announced. When will daily ticket sales peak and how many tickets will be sold that day?
for any quadratic function
f(x) = ax^2 + bx + c
the x of the vertex is -b/(2a)
sub that x value into the function to get the y of the vertex, so
n(x) = -.3x^2 + 9x + 15
x of vertex is -9/(2(-.3)) = 15
n(15 = -.3(225) + 9(15) + 15
= 82.5
ticket sales will peak on day 15 and they will sell 83 tickets, ( can't sell partial tickets)
check for reasonableness of answer
let x = 14
n(14) -.3(196) + 9(14) + 15 = 82.2 , < 82.5
let x = 16
n(16) = -.3(256) + 9(16) + 15 = 82.2 , < 82.5
I'll go with my answer
To find when daily ticket sales will peak and how many tickets will be sold on that day, we need to determine the maximum value of the function N(x)=-0.3x^2+9x+15.
To find the maximum value, we can use the vertex formula, which states that the x-coordinate of the vertex of a quadratic function in the form ax^2 + bx + c is given by x = -b / (2a).
In our case, a = -0.3 and b = 9. Plugging these values into the formula gives us:
x = -(9) / (2 * -0.3)
x = -9 / -0.6
x = 15
Therefore, the peak in daily ticket sales will occur 15 days after the concert was first announced.
To find the number of tickets sold on that day, we can substitute the value of x (= 15) into the function N(x):
N(15) = -0.3(15)^2 + 9(15) + 15
N(15) = -0.3(225) + 135 + 15
N(15) = -67.5 + 135 + 15
N(15) = 82.5 + 15
N(15) = 97.5
Therefore, on the 15th day after the concert was announced, approximately 97.5 tickets will be sold. Since you cannot sell a fractional part of a ticket, the actual number of tickets sold on that day will be either 97 or 98, depending on the rounding convention.
Hence, the daily ticket sales will peak on the 15th day, and approximately 97 or 98 tickets will be sold on that day.