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please answer all questions Camoative Box Office Revease 6mille) Movie Box Office Comparison PARTI: Modelling a...

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Camoative Box Office Revease 6mille) Movie Box Office Comparison PARTI: Modelling a curve 800 It is not easy to determine the
Camoative Box Office Revease 6mille) Movie Box Office Comparison PARTI: Modelling a curve 800 It is not easy to determine the best mathematical model for real data. In many situations, one model works best for a limited period of time, and then another model is better. Let x represent the time, in weeks, and let y represent the cumulative box office revenue, in millions of dollars. 700 600 500 400 Avater Ttenic 300 200 Derk Knight 100 0 10 15 20 25 Time (weeks) The curves for Avatar and Dark Knight appear to have a horizontal asymptote. What do you think this represents in this context? Do you think the curve for Titanic will eventually exhibit this characteristic as well? Explain. Consider the curve for Titanic. If the vertex is located at (22, 573), determine a quadratic function of the form y a(x - h)2+k that might model this portion of the curve. Suppose that the curve has a horizontal asymptote with equation y 600. Determine an exponential function of the formy= -35(0.65)03-)+k that might model the curve. Which type of function do you think better models this curve? Explain. earch acer
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Part 1.) The horizontal coordinate in the case of The Dark Knight and Avatar both represent the constant cumulative box office revenue that both these movies will have obtained once they are out of all the box offices. Then, since no new screenings will be done, the cumulative revenue stays a constant and thus, with time, a horizontal asymptote is achieved.

Once we reach a time when the box offices feel it is no longer profitable to keep screening Titanic, it too shall approach a horizontal asymptote.


Part 2.)
a.)

At the vertex, for any quadratic function, we have the slope to be zero, and this leads us to :

2a (22 h)0h 22 (a 0)

Also, since the vertex would lie on the curve itself, it is a solution to the given equation :

573 a(22 22)2 k k= 573

Further, we want (0,0) to also be a point on the curve, since at the start of time (t = 0 weeks), a movie cannot have non-zero cumulative box office revenue, that is y(0) = 0:

573 0 a(0 22)2 573 484

Hence, the curve is given by : 573 (r 22)2573 484 y =

b.)

Since y = 600 is a horizontal asymptote to the given curve we want to fit, it must be true that :

lim y600 エ→0 lm 35(0.65)0.3(-h) k) =600 エ→0 lim (35(0.65).3(r-h))lim k 600 (as xoo, h, Vh E R) エ→O0 エ→0 0,3r lim (35(0.65).)

Again, we would want (0,0) to be a point on the curve for the same reasons, which leads us to :

-0.3h 0 35(0.65).30-)60035(0.65)0 +600 600 -0.3h (0.65) 35 600 -0.3h) log10(0.65) log10 35 600 log10 35 h -0.3 log10 (0.65) h

Which leads us to the following equation :

y35(0.65)0.3(x-21.9877) +600

c.) It is evident that the exponential curve with the horizontal asymptote would fit the curve better, since it would fit the observed behavior so far well, as well as the long term (asymptotic) behavior as well. On the other hand, the quadratic function will curve downwards with increase in time (x), and thus not model the behavior of the cumulative box office revenue function (y) accurately.

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