Let's analyze the given function and the provided data in the table.
The function provided is:
[tex]\[ h(x) = -16x^2 + 100 \][/tex]
To determine for which value of [tex]\( x \)[/tex] this model makes the least sense to use, we can evaluate the height at different given [tex]\( x \)[/tex] values, which are:
[tex]\[
-2.75, \quad 0.25, \quad 1.75, \quad 2.25
\][/tex]
Here is the step-by-step evaluation of [tex]\( h(x) \)[/tex] for each given [tex]\( x \)[/tex]:
1. Evaluate [tex]\( h(-2.75) \)[/tex]:
[tex]\[ h(-2.75) = -16(-2.75)^2 + 100 = -16(7.5625) + 100 = -121 + 100 = -21 \][/tex]
2. Evaluate [tex]\( h(0.25) \)[/tex]:
[tex]\[ h(0.25) = -16(0.25)^2 + 100 = -16(0.0625) + 100 = -1 + 100 = 99 \][/tex]
3. Evaluate [tex]\( h(1.75) \)[/tex]:
[tex]\[ h(1.75) = -16(1.75)^2 + 100 = -16(3.0625) + 100 = -49 + 100 = 51 \][/tex]
4. Evaluate [tex]\( h(2.25) \)[/tex]:
[tex]\[ h(2.25) = -16(2.25)^2 + 100 = -16(5.0625) + 100 = -81 + 100 = 19 \][/tex]
Now, let's compare the results we have:
- [tex]\( h(-2.75) = -21 \)[/tex]
- [tex]\( h(0.25) = 99 \)[/tex]
- [tex]\( h(1.75) = 51 \)[/tex]
- [tex]\( h(2.25) = 19 \)[/tex]
The value [tex]\( h(-2.75) \)[/tex] results in a negative height, which doesn't make physical sense because height cannot be negative when representing the height of an object dropped from a certain point.
Thus, the model makes the least sense to use at [tex]\( x = -2.75 \)[/tex].
Therefore, for [tex]\( x \)[/tex], the value for which this model makes the least sense to use is:
[tex]\[ -2.75 \][/tex]
