Questions: The function f(x)=14.5-0.25 x^2 is used to model the curve of a train tunnel, where x is the distance from the center of the tunnel, and the x-axis represents the ground. The mathematical range for the function is the set of real numbers less than or equal to 14.5. Which statement describes how the reasonable range differs from the mathematical range? The reasonable range is the set of positive real numbers. The reasonable range is the set of positive whole numbers. The reasonable range is the set of real numbers between 0 and 14.5 inclusive. The reasonable range is the set of whole numbers between 0 and 14.5 inclusive.

The function f(x)=14.5-0.25 x^2 is used to model the curve of a train tunnel, where x is the distance from the center of the tunnel, and the x-axis represents the ground. The mathematical range for the function is the set of real numbers less than or equal to 14.5.

Which statement describes how the reasonable range differs from the mathematical range? The reasonable range is the set of positive real numbers. The reasonable range is the set of positive whole numbers. The reasonable range is the set of real numbers between 0 and 14.5 inclusive. The reasonable range is the set of whole numbers between 0 and 14.5 inclusive.

Transcript text: The function $f(x)=14.5-0.25 x^{2}$ is used to model the curve of a train tunnel, where $x$ is the distance from the center of the tunnel, and the $x$-axis represents the ground. The mathematical range for the function is the set of real numbers less than or equal to 14.5 . Which statement describes how the reasonable range differs from the mathematical range? The reasonable range is the set of positive real numbers. The reasonable range is the set of positive whole numbers. The reasonable range is the set of real numbers between 0 and 14.5 inclusive. The reasonable range is the set of whole numbers between 0 and 14.5 inclusive.

Solution

Solution Steps

Step 1: Analyze the mathematical range of the function

The function given is $ f(x) = 14.5 - 0.25x^{2} $. The mathematical range of this function is all real numbers less than or equal to 14.5 because the term $ -0.25x^{2} $ ensures that the function's output decreases as $ x $ moves away from 0.

Step 2: Consider the real-world context

In the real-world context of a train tunnel, the height of the tunnel cannot be negative. Therefore, the reasonable range must exclude negative values. This means the reasonable range is limited to non-negative real numbers.

Step 3: Compare the mathematical range with the reasonable range

The mathematical range includes all real numbers less than or equal to 14.5, which could theoretically include negative values. However, the reasonable range is constrained by the physical reality of the tunnel's height, so it must be the set of real numbers between 0 and 14.5 inclusive.

Step 4: Evaluate the given options

The reasonable range is the set of positive real numbers. (This is partially correct but does not specify the upper limit.)
The reasonable range is the set of positive whole numbers. (This is incorrect because the range includes all real numbers, not just whole numbers.)
The reasonable range is the set of real numbers between 0 and 14.5 inclusive. (This is correct as it aligns with the physical constraints.)
The reasonable range is the set of whole numbers between 0 and 14.5 inclusive. (This is incorrect because the range includes all real numbers, not just whole numbers.)

The correct statement is: The reasonable range is the set of real numbers between 0 and 14.5 inclusive.