Applied Mathematics for Business and Economics

Lecture Note Function

next 1200 cubic feet, and $50 per 100 cubic feet there after. Express the monthly
water bill for a family of four as a function of the amount of water used.

Solution
Let x denote the number of hundred-cubic-feet units of water used by the family
during the month andCx()the corresponding cost in dollars. If 0 ≤x≤ 12 , the cost is

simply the cost per unit times the number of units used.
Cx( )=1.22x

x 12

If 12 each of the first 12 units cost $1.22, and so the total cost of these 12 units is
1.22(12) = 14.64 dollars. Each of the remaining

x≤≤ 24

− units costs $10, and hence the total

cost of these units is (^10) (x− (^12) )dollars. The cost of all x units is the sum.
()=+−=−14.64 10(x^12 )^10 x105.
x 24
Cx
If , the cost of the first 12 units is 1.22 (12) = 14.64 dollars, the cost of the next 12
units is 10 (12) = 120, and the cost of the remaining
x> 24
− units is so (x− (^24) )dollars. The
cost of all x units is the sum.
Cx()=++−=−14.64 120 50(x (^24) ) 50 x1, 065.
Combining these three formulas, you get.

()

1.22 , if 0 12

10 105.36 if 12 24

50 1, 065.36 if 24

xx

Cx x x

xx

⎧ ≤≤

⎪

=−⎨ ≤≤

⎪ −>

⎩

The graph of this function

x (^) C(x)
0
12
24
30
0
14.
134.
434.
C(x)
450 -
400 -
350 -
0 -
6 12 18 24 30 X
300 -
250 -
200 -
150 -
100 -
5

4.3 Break-Even Analysis ................................................................................

Intersections of graphs arise in business in the context of break-even analysis. In a typical
situation, a manufacturer wishes to determine how many units of a certain commodity have to
be sold for total revenue to equal total cost. Suppose x denotes the number of units
manufactured and sold, and letCx()andR(x)be the corresponding total cost and total

revenue, respectively. A pair of cost and revenue curves is sketched in Figure: