Question 350710: The number of tickets sold each day for an upcoming performance of Handel's Messiah is given by N (x)= -0.5x^2+10x+14, where x is the number of days since the concert was first announced. When will daily tickets sales peak and how many tickets will be sold that day? Tickets sales will peak how many days after the concert was first announced?
Answer by haileytucki(390)   (Show Source):
You can put this solution on YOUR website! I just had a very similar problem to solve and am going to post it for you:
that's how I learned how to do this exact problem. The only thing that is different within you problem is a couple of the variables.
ax^2+bx+c=0 is the equation that we will be using for this problem.
Suppose you are an event coordinator for a large performance theater. One of the hottest new Broadway musicals has started to tour and your city is the first stop on the tour. You need to supply information about projected ticket sales to the box office manager. The box office manager uses this information to anticipate staffing needs until the tickets sell out. You provide the manager with a quadratic equation that models the expected number of ticket sales for each day x. ( is the day tickets go on sale).
a. Does the graph of this equation open up or down? How did you determine this?
DOWN
because the coefficient of the highest power term is negative
b. Describe what happens to the tickets sales as time passes.
As time goes by, the ticket sales will continue to decrease.
The way I determined this was by working out the equation as seen below:
x=-0.2(1)^(2)+12x+11
Expand the exponent (2) to the expression.
x=(-0.2*1)+12x+11
Multiply -0.2 by 1 to get -0.2.
x=(-0.2)+12x+11
Remove the parentheses around the expression -0.2.
x=-0.2+12x+11
Add 11 to -0.2 to get 10.8.
x=10.8+12x
Since 12x contains the variable to solve for, move it to the left-hand side of the equation by subtracting 12x from both sides.
x-12x=10.8
Since x and -12x are like terms, add -12x to x to get -11x.
-11x=10.8
Divide each term in the equation by -11.
-(11x)/(-11)=(10.8)/(-11)
Simplify the left-hand side of the equation by canceling the common factors.
x=(10.8)/(-11)
Simplify the right-hand side of the equation by simplifying each term.
x=-0.98
As said within the original equation if x=1 (day the sales begin), over time
As the solution depicts, the sales will decrease over time, eventually to not selling any at all.
C. 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.
Solve E(x) = 0 for "x>0"
x=(-12+-~(144-4*-0.2*11))/(-0.4)
Move the minus sign from the denominator to the front of the expression.
x=-((-12-~(144-4*-0.2*11))/(0.4))
Multiply -4 by -0.2 to get 0.8.
x=-((-12-~(144+0.8*11))/(0.4))
Multiply 0.8 by 11 to get 8.8.
x=-((-12-~(144+8.8))/(0.4))
Add 8.8 to 144 to get 152.8.
x=-((-12-~(152.8))/(0.4))
Take the square root of 152.8 to get 12.36.
x=-((-12-12.36)/(0.4))
Subtract 12.36 from -12 to get -24.36.
x=-((-24.36)/(0.4))
Move the minus sign from the numerator to the front of the expression.
x=-(-(24.36)/(0.4))
Reduce the expression -(24.36)/(0.4) by removing a factor of from the numerator and denominator.
x=-(-60.9)
Multiply -1 by each term inside the parentheses.
x=60.9, rounded up to 61 days
This shows that on the 61st day the ticket sales will cease, end at zero.
D. Will tickets peak or be at a low during the middle of the sale? How do you know?
Due to the set of points that are equally distant from the focus and the directrix, the parabola, has a maximum point, the tickets will peak during the middle of the sale.
E. . After how many days will the peak or low occur?
-b/2a=-12/-2*.2=120/4=30
On the 30th day, the peak will occur, as given in the above equation.
F. How many tickets will be sold on the day when the peak or low occurs?
E(30) = 191 tickets will be sold on the peak.
F. What is the point of the vertex? How does this number relate to your answers in parts e. and f?
-b/2a is the x of the vertex and the axis of symmetry
Y=-0.2(x-30)^2+191
Vertex is 30 and 191
G. How many solutions are there to the equation ? How do you know?
There are two solutions to this equations; it is a quadratic equation. As mentioned above, this polynomial equation (of the second degree) has a general form of ax^2 +bx+x=0, where a≠0.
H.What do the solutions represent? Is there a solution that does not make sense? If so, in what ways does the solution not make sense?
The solutions represent when “t” (ticket sales) were zero. The only solution that I see which does not make general sense is when Simplifing the right-hand side of the equation by x=-0.98. If that is a true statement then one would have to conclude that the ticket sales began almost a day earlier.
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