2.2.2 A Closer Look at the Model and the Solution

Since this is a real-world problem, it is not sufficient to check to see if our answer satisfies the equations, we must also verify that it is a reasonable answer to the original problem. Since the time correction, D, is related to the number of time zones crossed, our computed value means that Professor Tapia will cross time zones. But the time changes by one hour as soon as we cross a time zone boundary and then remains the same until we cross the next boundary. So it doesn't make much sense to talk about fractions of a time zone. Using the same reasoning, we see that it is not the case that the time zone correction increases by a small amount when the plane travels a small distance. Instead, the time zone correction is constant as long as the plane remains in a single time zone and then jumps abruptly as soon as the plane crosses into the next time zone. The only values of D that make sense are 0, 1, 2, 3, etc., and the value of D will be the same as the number of time zone boundaries crossed. Since we computed the value , it is reasonable to suspect that the true value of D is 5 or 6.

Now let's consider the variable T. Since we have used the same variable in both equations, we have assumed that the travel time in both directions is the same. In reality, it takes longer to travel from Geneva to New York (east to west) than from New York to Geneva (west to east).^† Professor Tapia chose to make the simplifying assumption that the travel times for both legs of the trip are the same.

Scientists often make this kind of assumption because it makes the problem easier (or possible) to solve. Sometimes the “real” problem is too difficult because the additional information needed to formulate it correctly is too complicated or is not available. Whenever we make a simplifying assumption, however, we must consider the effect that simplifying the model has on the solution. In this case, Professor Tapia felt that he could ignore the difference in travel time because he expected it to be small compared to the total travel time.

Professor Tapia initially built a model that didn't assume equal travel time. He defined T₁ to be the time in flight from New York to Geneva, T₂ to be the time in flight from Geneva to New York, and D to be the correction for time zones crossed. Then the equations for the trip out and the trip home are

(2.3)

It seems plausible that the difference in travel times should be proportional to the number of time zone boundaries crossed. Recall that D has units of time and is increased by one hour as soon as a time zone boundary is crossed. So a reasonable form for the third equation is

Now we have three conditions relating our three variables, but we can't use them to determine an answer to our problem until we have a value for a. This constant depends on factors such as prevailing winds and the rotation of the earth, and it will be difficult to calculate even if that data is known. So Professor Tapia made the reasonable assumption that a=0, which is equivalent to assuming T₁ = T₂ and leads to the simplified model (2.2).

Recall that D, the time correction, is determined by the number of time zone boundaries crossed and must be a whole number. The simple model provided the value , suggesting that D is either 5 or 6. Using the fact that the trip out actually takes less time than the trip home, Professor Tapia knew that T₁ must be less than T₂. Letting D = 5 contradicts this fact, so he decided that D = 6. This happens to be the true value for D. Professor Tapia set D =6 in each equation in (2.3) and saw that the time in flight going to Geneva is 7 hours and 40 minutes and the time in flight returning is 8 hours and 25 minutes.

We were unable to specify a value for the constant a in (2.4), so we assumed that a = 0. Now that we have values for T₁, T₂, and D, we can substitute this information into Equation (2.4) and compute that the corresponding value for a is . But we have no way to check that this is the true value. Without additional information, we cannot verify that D = 6 is the correct answer; but it certainly seems reasonable, and it is the answer that Professor Tapia used.

† Due to the rotation of the earth and prevailing winds.