Final answer:
The question deals with directly proportional relationships and finding the width of a rectangle given its length and proportionality constant. The constant of variation 'k' is derived from the given pairs and then used to calculate the width. However, the calculation does not match the provided options, suggesting an error in the question.
Step-by-step explanation:
The question involves the concept of directly proportional relationships, where two values or quantities increase or decrease together at the same rate. In mathematical terms, this is expressed with the equation y = kx, where y is the dependent variable, x is the independent variable, and k is the constant of proportionality.
To solve the given problem, we need to determine the width of a rectangle when we know its length is 27 centimeters and we are given options for the possible width measurements. To do this, we use the constant of variation k, which is the ratio of the length to the width. We can find this constant by looking at known ratios from the question. For example, a known pair (18, 162) implies that 18/162 is equal to k. Simplifying this ratio gives us 1/9, which means k = 1/9 and the relationship between the length L and width W of the rectangle can be defined as L = kW.
Now we can use k to find the width when L is 27 centimeters:
- L = kW
- 27 = (1/9)W
- W = 27 * 9
- W = 243
However, since 243 is not one of the given options for the width, there appears to be a mistake in provided options or in the original question's values. Therefore, based on the proportionality provided, we cannot determine the width from the options given.