Question

The number of tickets sold each day for an upcoming performance of​ Handel's Messiah is given N=-0.4x^2+9.6x+10,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?

Ticket sales will peak x days after the concert was first announced.

Answer 1

To find the day of peak ticket sales, we need to find the maximum value of N.

First, let's rewrite the equation in standard form:

N = -0.4x^2 + 9.6x + 10

N = -0.4(x^2 - 24x - 25)

N = -0.4(x - 12)^2 + 122

Now we can see that the maximum value of N occurs when (x - 12)^2 is minimized, which happens when x = 12.

Therefore, ticket sales will peak 12 days after the concert was first announced.

To find the number of tickets sold that day, we substitute x = 12 into the equation for N:

N = -0.4(12)^2 + 9.6(12) + 10

N = 70.4

Therefore, 70.4 tickets will be sold on the day of peak ticket sales. However, since you can't sell a fractional part of a ticket, the actual number of tickets sold will likely be rounded to the nearest whole number.

Answer 2

To find when daily ticket sales peak, we need to find the maximum point of the quadratic equation N = -0.4x^2 + 9.6x + 10.

One way to do this is to use the formula for the x-coordinate of the vertex of a quadratic function, which is:

x = -b/2a

where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c. In this case, a = -0.4 and b = 9.6, so:

x = -9.6 / 2(-0.4) = 12

This means that daily ticket sales will peak 12 days after the concert was first announced.

To find how many tickets will be sold that day, we can substitute x = 12 into the equation for N:

N = -0.4x^2 + 9.6x + 10 = -0.4(12)^2 + 9.6(12) + 10 ≈ 51.2

Therefore, approximately 51 tickets will be sold on the day that daily ticket sales peak, 12 days after the concert was first announced.

Explanation: