Section 1.10Modeling with Functions

Construct Functions from Verbal Descriptions

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.

Example

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.

1=70-.0025 70 xy 2 60 .005 60 xy

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

2

2

Any function in the form f(x)=ax

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.

bx c

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.

9”

14”

xx

Modeling the Area of a Rectangle Given a Specific Perimeter

Possible Lengths and Widths if the Perimeter is 140 feet

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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.

% amount Interest

Money market

CD

Continued on the next slide

Figure 1.80

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

2 3y x

x

y

(x,y)

(a) I=5(2000+x)+3x

(b) I=.05(2000+x)+.03x

(c) I=.05(2000-x)+.03x

(d) I=5(2000-x)+3x

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)

(b)

(c)

(d)

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=x(70-x)

A=x(400-x)

A=x(400+x)

A=x(200-x)