All of the scenarios you mentioned can involve irrational numbers at some point, but the one most likely to involve an irrational number in its calculations is finding the width of a rectangle based on the area and the length.
This is because the formula for the area \( A \) of a rectangle is given by \( A = \text{length} \times \text{width} \). If you have the area and the length, you would rearrange this formula to find the width:
\[ \text{width} = \frac{A}{\text{length}} \]
If the area is an irrational number and the length is a rational number, the resulting width can also be irrational.
For the other examples:
- Finding the total cost of filling up a car's tank will typically involve rational numbers (e.g., price per gallon).
- Finding the diameter of a circle based on the radius involves using the formula \( d = 2r \). If the radius is irrational, the diameter will also be irrational.
- Finding the side length of a square based on the area involves taking the square root of the area, which can also be irrational if the area is not a perfect square.
However, the rectangle scenario stands out as it is more common to encounter an area that is not a perfect square. Thus, it is particularly likely to yield an irrational width in practical real-world applications.