Question

The predicted equation for total sales over time is given by the equation: Total = 120*ln(t^2 + 0.63) + 50.7 where t is given in months and "Total" is given in thousands of units. Each copy of the movie sells for $14 and costs $7.7 to produce. The sunk cost of producing the film that has not been recovered from theater sales is $5100000
a.) Develop a general, linear profit equation, assuming the cost is linear and using x to represent a single unit of sales. Profit =
b.) Find an equation that represents total profit over time by using a composition of functions to change from x units to time

Answer 1

Profit is revenue minus cost.
Let \(x\) be the quantity sold.
Revenue in terms of quantity is given by: \[
r(x) = 14x
\]The cost in terms of quantity sold is given by:\[
c(x) = 7.7x+5100000
\]This means profit would be\[
p(x) = r(x) - c(x) = 14x-(7.7x+5100000)=6.3x-5100000
\]

Answer 2

Now, to find this in terms of time, remember that: \[
x(t) = 120\ln(t^2 + 0.63) + 50.7
\]

Answer 3

Thank you Wio, however for question a, we are looking for "x" to be a single unit of sales

Answer 4

Yes, which is why \(p(x)\) would be profit in terms of sales. Since \(x\) is number of sales.

Answer 5

Oh, ok...

Answer 6

Where as profit in terms of time is just \[\begin{split}
(p\circ x)(t) &= p(120\ln(t^2 + 0.63) + 50.7)\\
&=6.3(120\ln(t^2 + 0.63) + 50.7)−5100000\\
&=756\ln(t^2+0.63)-5099680.59

\end{split}\]

Answer 7

Thank you Wio. That is really helpful. So, for break even quantity would I just set 0=6.3x−5100000 and solve for x correct?

Answer 8

Yes.

Answer 9

ok great. I got 809523.809524 for break even qty. How long will it take in months to reach this quantity. How would I set up the equation?

Answer 10

Still there Wio?