3.3 Applications of Linear Functions
A function f is a linear function if
The graph of a linear function is a line with slope m and y-intercept b. The rate of change of a
linear function is the slope m.
Example 1: Graph
2
3
Solution: The slope of the line is 2 and the y-intercept is 3.
Most of this section deals with the following basic relationships:
Revenue = (price per item)(number of items) or
Cost = variable costs + fixed costs
Profit = Revenue – Cost or
The relationships in this section are linear functions
.
In examples 2-4, determine the following cost functions.
2. A gym charges a $175 initiation fee, plus $35 per month.
Solution: Cost = variable costs + fixed costs
35
175
3. The fixed cost is $2000 and 40 items cost $5000 to produce.
Solution:
2000
5000
40
2000
3000 40
75
75
2000
The cost function is
In economics, the rate of change of the cost function is called the marginal cost. If the cost
function is a linear function
, then the marginal cost is the slope m.
4. The marginal cost is $120 and 100 items cost $15,800 to produce, find the cost function.
Solution:
120
15,800 120 100
15,800 12000
3,800 = b
The cost function is:
120
3800
If
is the total cost in producing x items, then the average cost per item is given by:
5. If the total cost of producing x items is given by
average cost per item when 75,000 items are produced.
Solution:
75,000
8.75 75,000
16,500
$672,750
8.75
The average cost per item when 75,000 are items are produced is
16,500, determine the
,
,
$8.97
6. A machine is now worth $120,000 and will depreciate linearly over an 8-year period at which
time it will be worth $25,000 as scrap.
a) Determine the linear depreciation function f .
b) What is the domain of f ?
c) What will the machine be worth in 6 years ?
Solution: a)
120,000
25,000 8
120,000
95,000 8
11,875
11,875
120,000
b) The domain is 0,8
c) 6
11,875 6
120,000
6
$48,750
7. A computer that cost $1200 new is expected to depreciate linearly at a rate of $300 per year.
a) Determine the linear depreciation function f .
b) Explain why the domain of f is 0,4
Solution: a)
300
1200
b) The domain of f is 0,4 , since at 4 years the computer will be worth nothing and
after 4 years it would be worth a negative amount, which does not make sense.
8. The total cost (in dollars) of producing x compact disks is
6.80
a) What are the fixed costs?
b) What is the marginal cost per disk?
c) What is the total cost of producing 50,000 disks/
d) What is the average cost per item when 50,000 are produced?
Solution: a) $450,000
b) $6.80 per item
c) 50,000
6.80 50,000
450,000
50,000
$790,000
d)
,
450,000
$15.80
,
9. The manager of a restaurant found that the cost of producing 100 cups of coffee is $11.02,
while the cost of producing 400 cups is $40.12. Assume the cost function is linear.
a) Find the cost function
b) Find the total cost of producing 1000 cups.
c) What is the marginal cost?
d) What is the average cost per cup when 1000 cups are sold?
Solution: a)
11.02 100
40.12 400
Solve this system 11.02 100
40.12 400
29.1
300
.097
11.02 100 . 097
11.02 9.7
1.32
.097
1.32
b)
.097 1000
1.32
$98.32
c) The marginal cost is 9.7 cents per cup
d)
.
9.8 cents per cup
The number of units at which the revenue equals cost (that is profit equals 0) is the break-even
point.
10. The profit (in millions of dollars) from the sale of x million units of an item is given by
.7
25.5. The cost is given by
.9
25.5
a) Find the revenue equation
b) What is the revenue from selling 10 million units?
c) What is the break-even point?
Solution:
.7
25.5
.9
25.5
.7
25.5
.9
25.5
1.6
b)
1.6 10
R = $16 million
c) R = C
1.6
.9
25.5
.7
25.5
36.4 million items
11. The cost of producing x-items is given by
80
7000 and the revenue is given by
95 . Determine the break-even point. Should this item be produced if only 400 units can be
sold.
Solution: 80
7000 95
7000 = 15x
x = 466.666
The break even point is about 467 units. This item should not be produced since only
400 units can be sold, thus the company will not break even or make a profit.
The Law of Supply: As the price of a commodity increases the quantity supplied of that
commodity increases. This is because the business that makes that commodity wants to sell as
much of their good at the highest price.
The Law of Demand: An economic law that states that consumers buy more of a good when its
price decreases and less when its price increases.
In supply and demand problems, we use p for price and q for quantity. The equilibrium point is
the point where the supply curve intersects the demand curve.
12. Let the supply and demand for radial tires in dollars be given by
Supply:
Demand:
81
a) Graph these equations
b) Find the equilibrium quantity
c) Find the equilibrium price
Solution:
b)
81
3
3
4 81
2
4
6
324 3
9q = 324
q = 36
The quantity equilibrium is 36 units
4
c) The price equilibrium is
36
$54