Question

part iii: reflection

1. here is the graph of $f(x)=\frac{72}{x}$. describe the relationship between rate of return (x) and the number of years to double an investment (f(x)).

1. why is putting money into a savings account not the best way to save for retirement? use the rule of 72 to justify your response.

Explanation:

For question 6:

The function $f(x)=\frac{72}{x}$ is an inverse variation (hyperbolic) function. As the rate of return $x$ increases, the value of $\frac{72}{x}$ decreases, which matches the downward-sloping, decreasing curve shown. This means the two variables move in opposite directions.

For question 7:

Savings accounts have very low annual interest rates (typically well below 2%). Using the Rule of 72 ($f(x)=\frac{72}{x}$), a low rate leads to an extremely long time to double the investment. For example, with a 1% rate, it takes $\frac{72}{1}=72$ years to double, which is too slow to build sufficient retirement savings within a typical working lifetime.

Answer:

1. The rate of return ($x$) and the number of years to double an investment ($f(x)$) have an inverse relationship: as the rate of return increases, the number of years needed to double the investment decreases, and vice versa.
2. Savings accounts have very low annual interest rates. Using the Rule of 72, a low rate (e.g., 1%) results in a very long time to double savings ($\frac{72}{1}=72$ years), which is too slow to grow enough money for retirement in a reasonable timeframe, so it is not the best option.