How Are Inputs and Outputs Related Through A Model?

When working with a model, you need to pay careful attention to the units on each variable.

Problems 1 The number y (in millions) of women in the workforce is given by the function \displaystyle y=0.006{{x}^{2}}-0.018x+5.607 where x is the number of years after 1900.

a. Find the value of y when x = 44. Explain what this means.

b. Use the model to find the number of women in the workforce in 2010.

The solution above is the correct strategy, but there is an error…can you find the error?

How Do I Set Up and Solve a System with Inverses?

Writing a system of equations can be frustrating. In many cases, this starts when you do not write out which variables corresponds to what. How can you use “the smallest loan is one-half of the next larger loan” if you do not know which letter represents the amount of the  smaller loan and which letter represents the amount of the next larger loan?

Once you have the system, you can solve it with inverse matrices.

Problem 1 A bank gives three loans totaling 400,000 dollars to a development company for the purchase of three business properties. The largest loan is 100,000 dollars more than the sum of the other two, and the smallest loan is one-half of the next larger loan. Find the amount of each loan.

The key to writing out the equations for this problem is to make sure you know exactly which letter goes with which loan. Otherwise you don’t know whether to write x = 1/2y or y = 1/2x.

Once you have the solution (done with the inverse of A above), make sure it makes sense with the original problem statement. In the board below, the students solved the exact same problem using rref on their calculator. I expect that you will use some type of technology to do rref or find the inverse.

Problem 2 An investor has 400,000 dollars in three accounts, paying 6%, 8%, and 10%, respectively. If she has twice as much invested at 8% as she has at 6%, how much does she have invested in each account if she earns a total of 36,000 dollars in interest?

The second equation was originally y = 2x since the amount at 8% is twice the amount at 6%. This was then manipulated to put the system in a form where matrices can be used. Writing this equation out is MUCH simpler if you have written out what each variable represents somewhere (upper left) on the page.

How Can A Model Be Used To Answer Questions About Its Graph?

These problems each require you to utilize a graph to answer parts of the question. The difficulty is often gauging what the window should be. If you understand what the variables represent, your task is easier. For instance, in the first problem you know that the input has to do with years since 1995 and the outputs are percentages. This gives a lot of insight to an appropriate window.

Problem 1 The function \displaystyle P=5.8t+7.13 gives the percent of households with Internet access as a function of t, the number of years after 1995.

a. What are the values of t that correspond to the years 1996 and 2014?

b. P = f(10) gives the value of P for what year? What is f(10)?

c. What xmin and xmax should be used to set the viewing window so that t represents 1995-2015.

Where is the error in this solution?

Problem 2 The total cost of prizes and expenses of state lotteries is given by \displaystyle P=35{{t}^{2}}+740t+1207 million dollars, with t equal to the number of years after 1980.

a. What are the values of t that correspond to the years 1988, 2000, and 2012?

b. P = f(14) gives the value for P for what year? What is f(14)?

c. What xmin and xmax should be used to set a viewing window so that t represents 1980-2007?

Good work by these students! They actually did more than what the problem asked for.

Problem 3 A model that relates the median annual salary (in thousands of dollars) of females, F, and males, M, in the United States is given by \displaystyle F=0.78M-1.05.

a. Use a graphing utility to graph this function on a viewing window [0,100] by [0,80].

b. Use the graphing utility to find the median female salary that corresponds to a male salary of $63,000.

Many errors can be avoided by paying careful attention to the units on the variables (years since 1900, millions of women, thousands of dollars)! For instance, on the last problem a male salary of $63,000 corresponds to M = 63, NOT M = 63,000.

How Do You Find The Intercepts of a Linear Function?

Problem 2 The percent p of high school seniors who ever used marijuana can be related to x, the number of year after 2000, by the equation 25p + 21x = 1215.

a. Find the x intercepts of the graph of this function.

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

c. Graph the function using the intercept.