In the movie industry, the release date of a film is very important (this is true of any good, but let's consider movies for now
.) It is projected that a certain movie will do best in terms of total sales if released to DVD on Nov. 15. The predicted equation for total sales over time is given by the equation: Total = 100*ln(t^2 + 0.87) + 48.8 where t is given in months and "Total" is given in thousands of units. Each copy of the movie sells for $16 and costs $7.3 to produce. The sunk cost of producing the film that has not been recovered from theater sales is $5200000 .) Develop a general, linear profit equation, assuming the cost is lienar and using x to represent a single unit of sales. Answer: Profit =
b.) Find an equation that represents total profit over time by using a composition of functions to change from ”x” units to t (time). Be cautious of the units. Answer: Profit =
\ c.) Find the break even quantity of movies that must be sold. Answer:
d.) How long will it take (in months) until this quantity is reached? Answer: Answer(s) submitted
(c) Break even quantity is when p(x)-5200000=0, solving, 8.7x-5200000=0 => x=5200000/8.7 =597701.1 =597702 (rounded to next integer)
(d) Time t to reach break even is when P(t)=0 or solve for t in 870000*ln((t^2+0.87))-4775440=0 ln(t^2+0.87)=4775440/870000=5.489 raise powers to base of e t^2+0.87=e^(5.489)=242.018 t=sqrt(242.018-0.87)=15.53 months.
