Question

The Cozy Car Company created a new model of car especially for Japan and Vietnam, which they start shipping to both countries at the same time.
The company had shipped
\[2\] of that model of car to Japan by the end of the first month, and the cumulative total of cars they've shipped triples each month.
The company had shipped
\[5\] of that model of car to Vietnam by the end of the first month, and the cumulative total of cars they've shipped increases by
\[11\] cars each month.
The company's board assembles at the end of each month to review the cumulative total number of cars shipped to each country.
What is the first month in which the board will see the cumulative total number of cars shipped to Japan exceeded the cumulative total in Vietnam?

Answer 1

Let \( J_n \) represent the cumulative total of cars shipped to Japan at the end of month \( n \) and \( V_n \) represent the cumulative total of cars shipped to Vietnam at the end of month \( n \).

### Cumulative Total for Japan
- The cumulative total of cars shipped to Japan triples each month. Given that \( J_1 = 2 \), we can express the cumulative total for subsequent months as follows:
\[
J_1 = 2
\]
\[
J_2 = 3 \times J_1 = 3 \times 2 = 6
\]
\[
J_3 = 3 \times J_2 = 3 \times 6 = 18
\]
\[
J_4 = 3 \times J_3 = 3 \times 18 = 54
\]
\[
J_5 = 3 \times J_4 = 3 \times 54 = 162
\]

### Cumulative Total for Vietnam
- The cumulative total of cars shipped to Vietnam increases by 11 cars each month. Given that \( V_1 = 5 \), we can express the cumulative total for subsequent months as follows:
\[
V_1 = 5
\]
\[
V_2 = V_1 + 11 = 5 + 11 = 16
\]
\[
V_3 = V_2 + 11 = 16 + 11 = 27
\]
\[
V_4 = V_3 + 11 = 27 + 11 = 38
\]
\[
V_5 = V_4 + 11 = 38 + 11 = 49
\]

### Finding the First Month \( n \) where \( J_n > V_n \)
Now we will compare \( J_n \) and \( V_n \) for each month \( n \):

- For \( n=1 \):
\[
J_1 = 2, \quad V_1 = 5 \quad \Rightarrow \quad J_1 < V_1
\]

- For \( n=2 \):
\[
J_2 = 6, \quad V_2 = 16 \quad \Rightarrow \quad J_2 < V_2
\]

- For \( n=3 \):
\[
J_3 = 18, \quad V_3 = 27 \quad \Rightarrow \quad J_3 < V_3
\]

- For \( n=4 \):
\[
J_4 = 54, \quad V_4 = 38 \quad \Rightarrow \quad J_4 > V_4
\]

Thus, the first month in which the cumulative total of cars shipped to Japan exceeds the cumulative total shipped to Vietnam is:

\[
\boxed{4}
\]