A DVD producer has fixed costs (factory, machinery, and money paid to the recording artist) of $406,000. Additionally, for every DVD produced, the company spends $5 for materials and labor. The DVDs can be sold for $12 each.
a) Define a function, which gives the producer’s total cost in terms of the number of DVDs produced.
b) Define another function, which gives the total revenue in terms of the number of DVDs sold.
(c) Graph both the functions on the same set of axes.
(d) Locate the point of intersection.
(e) How many DVDs must they produce so that cost equals revenue (they breakeven)?
c = 406000 + 5n
r = 12n
at breakeven, c=r
406000+5n=12n
406000=7n
n = 58000
a) The producer’s total cost in terms of the number of DVDs produced can be defined by the following function:
Total cost = Fixed costs + (Variable cost per DVD * Number of DVDs produced)
In this case, the fixed costs are $406,000 and the variable cost per DVD is $5. The number of DVDs produced will be represented by the variable 'x'. Therefore, the function can be expressed as:
Total cost = 406,000 + (5 * x)
b) The total revenue in terms of the number of DVDs sold can be defined by the following function:
Total revenue = Price per DVD * Number of DVDs sold
In this case, the price per DVD is $12 and the number of DVDs sold will also be represented by the variable 'x'. Therefore, the function can be expressed as:
Total revenue = 12 * x
c) To graph both functions on the same set of axes, you can plot the number of DVDs produced or sold on the x-axis and the total cost or total revenue on the y-axis. The graph will have two separate lines, one representing the total cost and the other representing the total revenue.
d) To locate the point of intersection, you need to find the value of 'x' where the total cost and total revenue are equal. This can be done by setting the two functions equal to each other and solving for 'x'. The point of intersection will represent the breakeven point.
Total cost = Total revenue
406,000 + (5 * x) = 12 * x
Solving this equation will give you the value of 'x' at the point of intersection.
e) To find the number of DVDs they must produce to break even (where cost equals revenue), you need to solve the equation from part (d) to find the value of 'x'. This value will represent the number of DVDs they need to produce.