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Applications of functions in Business
Total Cost, Revenue, and Profit- The profit a firm makes in a product is the difference between the revenue
(amount received in sales) and its costs (amount spent by firm to produce the product). If x units are produced
and sold, it is written: P(x) = R(x)-C(x)
Where:
P(x) = profit from x sale of units
R(x) = total revenue from sale of x units
C(x) = total cost of production and sale of x units
-In general, revenue is found by the equation:
Revenue= (price of units) (number of units).
-The total cost C is composed of two parts, fixed costs and variable costs. Fixed cost, FC, remains
constant regardless of the number of units produced. Examples of fixed costs include depreciation, rent,
utilities, and so on. Variable costs, VC, are those directly related to the number of units produced. Thus,
the total cost is found by using the equation: C= FC + VC.
- A firm’s breakeven point occurs when P (x) = 0, or when R(x) = C(x).
Example:
A manufacturer sells a product for $10 per unit. The manufacturer’s fixed costs are $1200
per month, and the variable costs are $2.50 per unit. How many units must the
manufacturer produce each month to break even?
Solution:
The total revenue for x units of the product is R(x) =10x. The fixed costs are $1,200. The
variable cost is $2.50 for x units produced. Thus, the equation for total cost is
C(x)=2.50x+1200.
Using the equation:
R(x) = C(x)
10x = 2.50x+1200
7.5x =1200
x =160
Thus, the manufacturer will break even if 160 units are produced.
Supply, Demand and Market Equilibrium- the first quadrant of parabolas or other quadratic equations are
frequently used to represent supply and demand functions. Market equilibrium occurs when the quantity of units
demanded equals the quantity of units supplied. To solve for market equilibrium solve the system of equations
for quantity, q, and price, p.
Example:
If the supply function for a commodity is given by p = q 2 + 100 and the demand function
is given by p= -20q+2500 find the point of equilibrium.
Solution:
At the market equilibrium, both the equations will have the same
p-value. Thus substituting q 2 + 100 for p in p= -20q+2500 yields
q 2 + 100 = -20q+2500
q 2 +20q-2400=0
(q-40)(q+60)=0
q= 40 or q= -60
Because a negative quantity is not physically possible, the positive valued solution must
be used. The equilibrium point occurs when 40 units are sold, at (40,1700).
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