The table shows the approximate height of an object [tex]\( x \)[/tex] seconds after the object was dropped. The function [tex]\( h(x)=-16 x^2+100 \)[/tex] models the data in the table.

Height of a Dropped Object
\begin{tabular}{|c|c|}
\hline Time (seconds) & Height (feet) \\
\hline 0 & 100 \\
\hline 0.5 & 96 \\
\hline 1 & 84 \\
\hline 1.5 & 65 \\
\hline 2 & 37 \\
\hline
\end{tabular}

For which value of [tex]\( x \)[/tex] would this model make the least sense to use?

A. [tex]\(-2.75\)[/tex]

B. [tex]\(0.25\)[/tex]

C. [tex]\(1.75\)[/tex]

D. [tex]\(2.25\)[/tex]



Answer :

To determine which value of [tex]\( x \)[/tex] makes the least sense to use in the given model [tex]\( h(x) = -16x^2 + 100 \)[/tex], we should evaluate the height for each of the given [tex]\( x \)[/tex] values and check for physical validity.

Given the values to evaluate:
- [tex]\( x = -2.75 \)[/tex]
- [tex]\( x = 0.25 \)[/tex]
- [tex]\( x = 1.75 \)[/tex]
- [tex]\( x = 2.25 \)[/tex]

Let's evaluate the height for each of these values using the function [tex]\( h(x) = -16x^2 + 100 \)[/tex]:

1. For [tex]\( x = -2.75 \)[/tex]:
[tex]\[ h(-2.75) = -16(-2.75)^2 + 100 \][/tex]
[tex]\[ = -16 \cdot 7.5625 + 100 \][/tex]
[tex]\[ = -121 + 100 \][/tex]
[tex]\[ = -21 \][/tex]

2. For [tex]\( x = 0.25 \)[/tex]:
[tex]\[ h(0.25) = -16(0.25)^2 + 100 \][/tex]
[tex]\[ = -16 \cdot 0.0625 + 100 \][/tex]
[tex]\[ = -1 + 100 \][/tex]
[tex]\[ = 99 \][/tex]

3. For [tex]\( x = 1.75 \)[/tex]:
[tex]\[ h(1.75) = -16(1.75)^2 + 100 \][/tex]
[tex]\[ = -16 \cdot 3.0625 + 100 \][/tex]
[tex]\[ = -49 + 100 \][/tex]
[tex]\[ = 51 \][/tex]

4. For [tex]\( x = 2.25 \)[/tex]:
[tex]\[ h(2.25) = -16(2.25)^2 + 100 \][/tex]
[tex]\[ = -16 \cdot 5.0625 + 100 \][/tex]
[tex]\[ = -81 + 100 \][/tex]
[tex]\[ = 19 \][/tex]

From our evaluations, we observe that:

- For [tex]\( x = -2.75 \)[/tex], the height [tex]\( h(-2.75) \)[/tex] is -21, which is a non-physical, negative height.
- For [tex]\( x = 0.25 \)[/tex], the height [tex]\( h(0.25) \)[/tex] is 99, a valid height.
- For [tex]\( x = 1.75 \)[/tex], the height [tex]\( h(1.75) \)[/tex] is 51, a valid height.
- For [tex]\( x = 2.25 \)[/tex], the height [tex]\( h(2.25) \)[/tex] is 19, a valid height.

Since a negative height does not make physical sense in this context (as heights should be non-negative), we conclude that the value of [tex]\( x \)[/tex] for which the model makes the least sense to use is [tex]\( x = -2.75 \)[/tex].