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].
