Category: Finite Math

How Do You Find a Linear Model from a Rate

In another MathFAQ,  I examined how we can find the equation of a line from two data points. In this post I want to look at a closely related problem where we find the equation of the line from a rate.

Problem Assume the growth of the population of Del Webb’s Sun City Hilton Head community was linear from 1996 to 2000, with a population of 198 in 1996 and a rate of growth of 705 people per year.

a. Write an equation for the population P of the community where x is the number of years after 1990.

Solution The population of 198 in 1996 corresponds to the point (6, 198) since the variable x corresponds to years after 1990. We’ll write the slope-intercept form of the line, P = mx + b, and substitute m into the equation. The rate of growth, 705 people per year, is the slope of the function. Therefore, the line describing the population is

P = 705x + b

To find the value of b, we need to substitute x = 6 and P = 198:

198 = 705(6) + b

-4032 = b

This gives us the equation,

P = 705x – 4032

b. Use the function to estimate the population in 2002.

Solution The year 2002 corresponds to x = 12. Substitute this value into the function to yield

P = 705(12) – 4032 = 4428

The population in 2002 will be 4428 people.

c. In what year will the population reach 10,000?

Solution In this part, set P = 10,000 and solve for x.

10000 = 705x – 4032

14032 = 705x

14032/705 = x

19.9 ≈ x

This corresponds to 1990 + 19.9 = 2009.9. So in the year 2009, the population will reach 10000. Since we are asking in what year, we DO NOT round up on the answer.

How Do You Find the Equation of a Line From an Application?

Below are some problems that students solved on the board in previous semesters. All of them start from the slope-intercept form of a line and require you to find the slope m from two points and then solve for b. Click on the pictures to see a larger version.

Problem The percent p of adults who smoke cigarettes can be modeled by a linear equations p = mt + b, where t is the number of years after 1960. If two points on the graph of this function are (25, 30.7) and (50, 18.1), write the linear equation of this application.

Solution

 

Problem The number of women in the workforce, based on data and projections from 1950 to 2050, can be modeled by a linear equation y = mx + b. The number was 18.4 million in 1950 and is projected to be 81.6 million in 2030. Let x represent the number of years after 1950 and y be the number of women in the workforce in millions.

a. What is the slope of the line through (0, 18.4) and (80,81.6)?

b. What is the average rate of change in the number of women in the workforce during this time period?

c. Use the slope from part a and the number of million of women in the workforce in 1950 to write the equation of the line.

Solution

How Do You Interpret A Linear Model?

Linear model are not always given in a nice slope-intercept form. If they were, it would be easy to read the slope and y-intercept off the y = mx + b form. In the example below, you are given a linear model in general form, ax + by = c. However, the letters are p and x instead of x and y.

Problem The percent p of high school seniors using marijuana daily can be related to x, the number of years after 1990, by the equation 30p – 19x = 30.

a. Find the x-intercept of the graph of this function.

Solution To find an intercept, we need to set the other variable equal to zero. In this case, to find the x-intercept, we need to set p = 0:

b. Find and interpret the p-intercept of the graph of this function.

Solution To find the p-intercept, set x = 0:

Since x corresponds to years after 1990, this tells us that in 1990 the percentage of seniors using marijuana daily was 1%.

c. Graph this function, using the intercepts. What values of x on the graph represent the years 1990 and after?

Solution Put the two points we have found on a graph.

Since 1990 and beyond correspond to x values 0 and above, a better viewing window would be one like we see below.

Notice that with each axis labeled, it makes more sense to use non-negative x values ( 0 and greater).

d. Use this model to determine the slope of the graph of this function if x is the independent variable.

Solution We could use the points we found earlier to calculate the slope, but it is easier to change the equation into slope-intercept form and read off the slope. To do this, we’ll need to solve 30p – 19x = 30 for p:

Since this is in the slope-intercept form p = mx + b, the slope is 19/30 or approximately 0.63. The line we drew above also makes sense since it has a positive slope and a vertical intercept of 1.

e. What is the rate of change of the percent of high school seniors using marijuana each year?

Solution The rate of change is the slope, with the appropriate units attached. In this case, the units are

So the percent of students using marijuana is rising at a rate of 19/30 percent per year. Since 19/30 is approximately .63, this means that each year the percentage increased by about .63%.

Interpreting this model means we examined the intercepts and their meaning as well as the slope and what it tells us about the percent using marijuana and the years since 1990.

How Can You Use a Spreadsheet To Carry Out The Simplex Method?

The row operations in the Simplex algorithm are not difficult to compute.  To start, you should carry out the row operations in the process by hand. As you will quickly realize, the Simplex algorithm is tedious and prone to arithmetic errors. Once you get the idea behind how the Simplex method works, you can carry out the row operations in a spreadsheet.

In the video below, the steps in the Simplex Methods are carried out in Google Sheets. The same steps can also be carried out in other spreadsheets such as Microsoft Excel.

Please note that there is a typo in the original matrix in the process. The cell in D2 should be a 1 instead of a zero to account for the slack variable in the first constraint.

How Do You Solve A Standard Minimization Problem?

In an earlier FAQ, we wrote out a standard minimization problem from an application. Now we’ll solve that problem by finding the dual standard maximization problem and applying the Simplex Method.

This process is covered in Section 4.4 of the textbook. For complete details on the process, consult this section and the many examples contained in this section. I will assume that you have looked over this section and are familiar with carrying out row operations on a matrix.

The problem we will solve is

Minimize C = 16y1 + 14y2 + 12y3 subject to

y1 + y2 + y3 > 6000

2y2 – 3y3 > 0

3y1 – y2 – y3 > 0

y1 > 0, y2 > 0, y3 > 0

Solution Write out the dual problem and apply the Simplex Method. Start by entering the objective function and constraints into a matrix. The transpose of this matrix yields the form of the dual standard maximization problem.

The dual problem is

Maximize z = 6000x1 subject to

x1 + 3x3 < 16

x1 + 2x2 – x3 < 14

x1 – 3x2 – x3 < 12

x1  > 0, x2 > 0, x3 > 0

Now add the slack variables to each constraint and form the initial matrix for the Simplex Method.

The pivot column is the first column since that is where the indicator row is most negative. The pivot row is the third row. The entry in the pivot is already a zero so we simply need to put zeros in the rest of the column using row operations.

For this new matrix, the pivot is the 5 in the second row, second column. Change the pivot to a 1 and then use it to put zeros in the rest of the column. This results in

Since there is still a negative number in the indicator row, we continue applying the Simplex Method. In this step, the pivot is the 4 in the first row, third column. As before, this entry is changed to a 1 and used to make the rest of the column into zeros.

The indicator row does not contain any negatives numbers so this is the final matrix. The solution to the standard minimization problem is under the slack variables and indicate that

y1 = 1500, y2 = 2700, and y3 = 1800

with a corresponding minimum of C = 83400 cents or $834.