Section 1.10
Modeling with Functions
Construct Functions from
Verbal Descriptions
Example
Write the function that will solve this problem.
Spice Drops candy calorie count exceeds Smarties
candy calorie count by 70 calories per serving. If the
sum of one serving of each candy equals 190 calories
find the calorie count of each kind of candy.
Continued on the next slide
Example
Example
Write the functions for men and
women and solve the problem.
The percentage of women in the labor
force and the percentage of men in
the labor force is illustrated in the
graph at left. The decrease yearly of
men in the labor force is ¼% and the
increase in women in the labor force
is ½%. If there are presently 70 million
men and 60 million women in the
labor force, when will the number of
both sexes be equal?
Graphing Calculator
Solving the previous problem using intersection. Let y1
be the left side of the equation and y2 be the right side of
the equation.
y 2  60  .005  60 x
y1=70-.0025 70 x
Example
A local telephone company charges $11 for local
phone service and an additional $ .10 for each long
distance phone call. A second local telephone
company charges $14 for local service and an
additional $ .05 for each long distance phone call.
For how many minutes of long-distance calls will the
costs for the two companies be the same?
Continued on the next slide
Number of Passengers Per Month
Revenue Per Month
Any function in the form f(x)=ax 2  bx  c
where a  0, is called a quadratic function.
In this chapter we use the U-shaped graph
of the stadard quadratic function, f(x)=x
to graph various transformations.
2
Functions from Formulas
Example
A machine produces open boxes using rectangular sheets of metal
measuring 14 inches by 9 inches. The machine cuts equal-sized
squares from each corner. Then it shapes the metal into an open box by
turning up the sides. Express the volume of the box, V, in cubic inches
as a function of the length of the side of the square cut from each
corner, x, in inches.
x
x
9”
14”
Modeling the Area of a Rectangle Given a Specific Perimeter
Possible Lengths and Widths if the Perimeter is 140 feet
Continued oon the next slide
Modeling the Area of a Rectangle Given a Specific Perimeter
Example
You have 300 feet of fencing to fence in a corral for your horse.
Express the area of the fenced in area as a function of one of its
dimensions.
?-x
Annual Simple Interest on an investment
The annual simple interest that an investment earns is given by the formula
I=Pr
where I is the simple interest, P is the principal, and r is the simple interest
rate expressed in decimal form. Suppose that you deposit $400 at 3% (r=0.03).
I=$400 x .03=$12
Example
A woman who was going to retire had $100,000 that she
invested in her local bank. She put some of the money in a
money market account at 3 ½% and some in a certificate of
deposit at 4%. If the first year’s interest is $3850, how much
money did she put in each account?
Write the functions you would use to solve this problem.
%
Money
market
CD
amount Interest
Figure 1.80
Continued on the next slide
Example
If a cylindrical can is to hold 50 cubic inches of oil. Write the surface
area of the cylinder in terms of “r.”
Example
Find the distance from the point on the parabola to the origin.
Express it as a function of x. The parabola can be described as
y  x2  3

y


(x,y)

 










x

If you invest a total of $2000 in two accounts.
One account pays 5% and another account
pays 3%. Write the equation that expresses
the amount of interest that you will make in 1
year in terms of x, the amount invested at 3%.
(a) I=5(2000+x)+3x
(b) I=.05(2000+x)+.03x
(c) I=.05(2000-x)+.03x
(d) I=5(2000-x)+3x
You have 400 meters of fencing to fence
in a garden. Write the formula for the area
of the garden as a function of x its width.
(a)
A=x(70-x)
A=x(400-x)
(c) A=x(400+x)
(d) A=x(200-x)
(b)