Focus Problem- Math

Question

The next page contains some information related to mpg (miles per gallon), vehicle efficiency, and governmental regulatory standards/penalties [note the CAFE standards penalty referenced are not current – although similar CAFE standards/penalties are still part of US law]. It also has some questions scattered throughout. For accuracy and consistency, generally round most numbers to (at least) 5 significant figures (5 numbers after the first non-zero number).

MPG vs GPM?!

Most Americans are familiar with and use mpg. Similarly, many of us are highly comfortable with linear models and linear projections. When we think of “average”, the default thinking is generally “sum up and divide by how many”… this is a mean, more specifically it is called the Arithmetic Mean. Unfortunately, being locked into this described way of thinking can lead to misconceptions, inefficiencies, and unintended consequences.

Situation 1 (warm-up): Alice, Bob, and Cindy each drive about 1000 miles per month. Alice drives a truck which gets 10 mpg. Bob drives a 20 mpg car. Cindy drives a smaller vehicle which averages 40 mpg. The goal is to find the 3 friend’s “average” mpg… Computing the regular arithmetic mean of the 3 mpg will lead to an incorrect average. Instead, there are two alternatives. One, invert each mpg to get gpm then compute the arithmetic mean, then re-invert back to mpg [this method is called a Harmonic Mean]. Two, find the total number of miles driven and the total number of gallons used, then divide. Execute one or both of these methods and compare/contrast to the incorrect arithmetic mean.

The three friends plan to buy newer, more efficient vehicles. Alice’s new truck will get 13mpg. Bob is upgrading to a 28mpg car. Cindy is looking at a 55mpg hybrid. Who will save more gasoline with these upgrades? What upgraded mpg would the other two have needed in order to save the same amount? Again, there are two ways to find the answers. One, compare the gpm for each person’s new car to their old car. Then find out how many gpm are saved by each person. Finally, take the largest saved gpm and apply that savings to the other two before inverting back. Two, compare the gallons used for each person’s new car to their old car (each month). Then take the largest saved number of gallons and apply to the other two. Execute one or both of these methods and summarize!

Situation 2 (main event!): Each month ZV Corp (Zebra Vehicles where all paint jobs are striped) makes 230 A vehicles which get 10mpg, 500 B vehicles which get 20mpg, and 70 C vehicles which get 40mpg. Unfortunately, the government sets a standard where companies must “average” 30mpg [this is called a CAFE law]. Any company that falls short of the 30mpg standard must pay a penalty. The penalty is calculated as $55 times number of vehicles sold each month times the deficit in their average mpg from the 30mpg standard.

• Compute ZV Corp’s monthly penalty if the government incorrectly uses (weighted) arithmetic mean to compute average mpg.

• Compute ZV Corp’s monthly penalty if the correct Harmonic Mean is used.

• ZV Corp is beginning to make a D vehicle which gets 75mpg. They estimate initially adding 2 D vehicles to their production each month. Compute their new monthly penalty.

Answer 1