The number of tickets sold each day for an upcoming performance of Handel's Messiah is given by N(x)= -0.4x^2 + 8.8x +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?
Ticket sales will peak ___ days after the concert was first announced.
The number of tickets sold on that day will be ____. (Round to the nearest integer)
find the roots to the quadratic. Maximum will be halfway between the roots.
That's where I got lost was when I was trying to find the roots of the quadratic equation. I guess I just need more practice with these roots.
0.4 x 0.4 = 0.16
8.8 x 8.8 = 77.44
15 x 15 = 225
to find the roots of -0.4x^2 + 8.8x +15 = 0
I would first "clean-up" the equation. I don't like decimals in my equations so I would multiply each term by 10
-4x^2 + 88x + 150 = 0
divide by -2
2x^2 - 44x - 75 = 0
a=2, b=-44, c=-75
x = (-b ± √(b^2 - 4ac))/(2a)
= (44 ± √2536)/4
= 23.5897 and -1.58987
the halfway value of x, as bobpursely suggested, is 11
so N(11) = -.4(121) + 8.8(11) + 15 = 63.4
another way is to know that for any function
f(x) = ax^2 + bx + c, the max/min occurs when
x = -b/(2a)
then -b/(2a) = -8.8/(2(-.4)) = 11 as above
So then I would end up with the ticket sales would peak 11 days after, and the number of tickets sold that day would be 63.4 Correct or No?
Although it says to round to the nearest integer. So this would actually be sales peak 11 days after, and the number of tickets sold would just be 63.
correct
Ahhhh Thank you very much! I seem to get lost when it comes to the cleaning up the decimals & then trying to get the square root if it is in decimal form. I appreciate your help Reiny :)!
The number of tickets sold each day for an upcoming Handel’s Messiah is given by N(x)=-0.5x^2+12x+15
To find out when daily ticket sales peak and the number of tickets sold that day, we need to determine the maximum value of the function N(x) = -0.4x^2 + 8.8x + 15.
The maximum value of a quadratic function occurs at its vertex, which is given by the formula x = -b/ (2a), where a, b, and c are coefficients of the quadratic equation ax^2 + bx + c = 0.
In this case, the coefficient of x^2 is -0.4, the coefficient of x is 8.8, and the constant term is 15. Plugging these values into the formula, we get:
x = -8.8 / (2*(-0.4)) = -8.8 / (-0.8) = 11.
The x-coordinate of the vertex is 11, which means the peak ticket sales will occur 11 days after the concert was first announced.
To find out the number of tickets sold on that day, we substitute x = 11 into the function N(x):
N(11) = -0.4(11)^2 + 8.8(11) + 15
= -0.4(121) + 8.8(11) + 15
= -48.4 + 96.8 + 15
= 63.4.
Therefore, the number of tickets sold on the day of the peak ticket sales will be approximately 63 (rounded to the nearest integer).