PreCalculusChapter 1 Section 7MODELING WITH FUNCTIONS

Functions from Formulas Write the area A of a circle as a function of its

A. Radius: rB. Diameter: dC. Circumference: C

Functions from Formulas Write the area A of a circle as a function of its

A. Radius: A(r) = π r2

B. Diameter: A(d) = π (d/2)2 = (π / 4)d2

Remember: r = d/2

C. Circumference: A(C ) = π(C/2π)2 = C2/(4π)Remember: C = 2 π r

Max Volume Problem A square of side x is cut out of each corner of an 8 in. by 15 in. piece of cardboard and the sides are folded up to form an open - topped box. See figure 1.80 on page 152.

A. Write the volume V of the box as a function of x.B. Find the domain of V(x).

Note the model imposes restrictions on x.C. Graph V(x) over the domain found in part (B) and use the maximum finder

on your grapher to determine the maximum volume of the box.D. How big should the cut-out squares be in order to produce the box of

maximum volume.

Max Volume Problem

A. The box will have a base with sides of width 8 – 2x and length 15 – 2x. Why?

The depth of the box will be x when the sides are folded up. Why?

Therefore, V(x) = x(8 – 2x)(15 – 2x)

Max Volume Problem

B. The formula for V is a polynomial with domain of all reals. However, we are restricted by the reality of the problem. We cannot cut a negative length. Also, we are restricted by how far into the cardboard we can cut. What side of the cardboard is smallest? And how far in can we cut?

We get a relevant domain of [0, 4]

Max Volume Problem

C. Look at the graph in the restricted domain of [0, 4]. Use the maximum finder on the calculator to determine the maximum occurs at the point (1.66…, 90.74) or (5/3, 90.47)

D. Each square should have sides of 5/3

Functions from Verbal Descriptions

In this method, finding the function is sometimes harder than solving the problem.

Be very careful when you read the problem and make sure you understand it.

Note: many times the rate of change in the problem is represented in different ways.◦ For example: the problem may describe the rotation of a tire in

inches per second but may want the answer in miles per hour.

Finding a Model Grain is leaking through a hole in a storage bin at a constant rate of 8 cubic inches per minute. The grain forms a cone-shaped pile on the ground below. As it grows, the height of cone always remains equal to its radius. If the cone is one foot tall now, how tall will it be in one hour?

What formula did you need to look up in order to solve this problem?

What information do we know that can be used in that formula?

Finding a Model Vcone = (1/3)π r2 h : However, we are told from the problem that the height is equal to the radius, and that we want to find the height after an hour.

So our function of Volume with respect to height is: Vcone = (1/3) π h3

When h = 12 inches, V = (1/3) π (12)3 =576π in3

An hour later, the volume will have grown by (60min)(8in3/min) = 480 in3

So, V = 576π +480 = (1/3)π h3 , solve for h.

inchesh 98.12)480576(3

3

Functions From Data Given a set of data points of the form (x, y), to construct a formula that approximates y as a function of x:

1. Make a scatter plot of the data points. The points do not need to pass the vertical line test.

2. Determine from the shape of the plot whether the points seem to follow the graph of a familiar type of function (line, parabola, cubic, sine, etc)

3. Transform a basic function of that type to fit the points as closely as possible. Do a regression analysis with a graphing calculator.

Finding Functions from Data

See Example 6 on page 156 for finding a regression line (line of best fit)

Modeling with Functions We will continue this idea through Chapter 4.

Please note page 157 in your text.◦ This page has the basic functions you will need to know and their

applications to “real world” problems.

Homework # 3 – 39 by multiples of 3 on page 160 & 161.