Module 7: Rational Functions and Function Operations

 

Rational equations can be used to model and solve some problems. Consider the German tank problem in Try This 4.

 

Try This 4

 

This photo is of a World War II tank.

iStockphoto/Thinkstock

In World War II, it was of interest to the Allies to know how many weapons, such as tanks, the Axis powers had produced. One method the Allies used to estimate this was to use the serial numbers of captured or destroyed weapons. For this strategy, the total number of serial numbers, n, and the largest serial number, m, of captured weapons were used. Two assumptions were then made using these serial numbers:

As an example, suppose that the Allies recovered tanks with engine serial numbers 7, 10, 14, 15, 21, and 24. The largest number found was 24, so assume that at least 24 tanks were produced. The average gap between serial numbers can be found by dividing the number of missing serial numbers by the number of serial numbers found, as follows:

 

 

 

This means there are approximately 3 more tanks than the 24 known for sure, so there were approximately 27 tanks produced.

 

This image shows 27 short vertical lines representing tanks. The lines 7, 10, 14, 15, 21, and 24 are labelled in blue with text stating that 6 serial numbers altogether were found. The label 24 is circled in red with text stating that 24 was the largest serial number found. Line 27 is labelled in green with text stating that 27 is the estimated number of tanks produced. Lines 25, 26, and 27 are grouped in purple with a 3 above the grouping with text stating that 3 is the estimated size of the last gap.

  1.  
    1. Explain why  represents the average gap of missing numbers.
    2. Using m and n as described, and T to represent the estimated total number of weapons produced, write a rational equation that can be used to estimate the total number of weapons produced.
    3. Use your equation to estimate the total number of weapons if 22 weapons are found and the largest serial number is 5217.
  2. Begin again with the formula you created in question 1.b.
    1. Set m as a constant at 1000. Graph the estimated total, T, as a function of the number of weapons found, n.
    2. Set n as a constant at 100. Graph the estimated total, T, as a function of the largest serial number found, m.
    3. Which variable, m or n, appears to have a greater effect on the estimate of weapons? Explain.
  3.  
    1. Suppose that the Axis powers found out what the Allies were doing and wanted to trick the Allies into thinking they had produced more weapons than they really had. If they were able to determine that the Allies had captured approximately 250 weapons of a particular type, and wanted them to believe they had 7000, what fake serial number should they let the Allies capture? Solve using a graphical method.
    2. How important is the estimate of 250 in determining the fake serial number in the question 3.a? Explain.
    3. Explain whether or not you think the Allies would have fallen for the trick.

course folder Save your responses in your course folder.

 

Share 3

 

With a partner or group, discuss the following questions based on the information from Try This 4.

  1. Explain any problems you see using the estimating procedure described in Try This 4.
  2. Describe a manufacturing method that could be used to make it harder to predict weapon numbers from serial numbers.

course folder If required, save a record of your discussion in your course folder.


The formula should have looked similar to