Question

The function f(x) = 14.5 – 0.25x2 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?

answer choices:
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.

Answer 1

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

The function $f(x) = 14.5 - 0.25x^2$ is used to model the curve of a train tunnel,

The graph of function will be inverted U shaped figure.

So the range will be less than y coordinate of vertex

Lets find the vertex

formula to find x co-ordinate of vertex is $x=\frac{-b}{2a}$

$f(x) = 14.5 - 0.25x^2$, a=-0.25, b=0

$x=\frac{-0}{2*(-0.25)}=0$

Now we plug in 0 for x  in f(x)

$f(x) = 14.5 - 0.25(0)^2= 14.5$

The y co-ordinate of vertex is 14.5

We know x axis represents the ground. The height of the curve cannot be negative. So, the curve of a tunnel starts at 0.  The range is from 0 to 14.5

We cannot take only whole numbers for y.

So, The reasonable range is the set of real numbers between 0 and 14.5 inclusive.