Given the problem, we need to determine which equation best models the provided data set [tex]\((x, y)\)[/tex].
The data points are:
[tex]\[ (0, 32), (1, 67), (2, 79), (3, 91), (4, 98), (5, 106), (6, 114), (7, 120), (8, 126), (9, 132) \][/tex]
Let's determine the linear relationship [tex]\( y = mx + b \)[/tex] where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept of the line that fits this data.
After performing the calculations, we obtain the values of the slope [tex]\( m \)[/tex] and the intercept [tex]\( b \)[/tex]:
[tex]\[ m \approx 9.67 \][/tex]
[tex]\[ b \approx 53 \][/tex]
Now, let's examine the given choices:
A. [tex]\( y = 33 \sqrt{x} + 32.7 \)[/tex]
This option suggests a square root relationship, which does not align with our linear findings.
B. [tex]\( y = 33 x - 32.7 \)[/tex]
This option does not match our calculated intercept or slope.
C. [tex]\( y = 33 \sqrt{x - 32.7} \)[/tex]
This also suggests a square root relationship and an incorrect form, not fitting a linear model.
D. [tex]\( y = 33 x + 32.7 \)[/tex]
This suggests a linear equation, but with a slope of 33 instead of 9.67 and a y-intercept of 32.7 instead of 53.
Comparing the calculated slope and intercept:
[tex]\[ \text{Slope} \approx 9.67\][/tex]
[tex]\[ \text{Intercept} \approx 53 \][/tex]
Choice D does have a linear form, but neither the slope (9.67 vs. 33) nor the intercept (53 vs. 32.7) match our findings.
Although none of the given choices match our calculated parameters exactly, if there was a typographical oversight in the problem where the given slope and intercept of D were intended to be closer to a typical linear model, the most likely typographical choice would be between option D. Given the set limitations and choice D's linear form, despite incorrect parameters, we note that it is the better approximation in form among the provided options.
Therefore, among the given choices:
[tex]\[
\boxed{D}
\][/tex]
