Question

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

Answer 1

(a) Profit equation
Profit = revenue - cost
unit profit = revenue per unit - cost per unit = 16-7.3=8.7
profit
p(x) = (unit profit)*x=8.7x

(b) profit over time
sales,
x(t) = 1000 * (100 * (ln(t^2+87)+48.8)
=100000 ln(t^2+0.87)+48800

profit,
P(t)
=p(x(t))-5200000
=p(100000 ln(t^2+0.87)+48800)-5200000
=8.7(100000 ln(t^2+0.87)+48800)-5200000
=870000*ln((t^2+0.87))-4775440

(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.