To develop a model for Miles' height, we need to find a square root function that fits the given data. The data for ages [tex]\(x\)[/tex] and heights [tex]\(y\)[/tex] is as follows:
| Age, [tex]\(x\)[/tex] | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 |
|--------------|----|----|----|----|----|----|----|----|----|
| Height, [tex]\(y\)[/tex]| 54 | 58 | 61.5 | 64.5 | 68 | 70 | 73 | 74 | 74.5 |
The square root function we are using is of the form:
[tex]\[ y = a \sqrt{x} + b \][/tex]
To find the parameters [tex]\(a\)[/tex] and [tex]\(b\)[/tex] that best fit the data, we use regression techniques. Through this process, we determine that the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are:
[tex]\[
a = 19.730588892721208
\][/tex]
[tex]\[
b = -7.116197572583441
\][/tex]
Therefore, the equation of the square root function that best models the given data is:
[tex]\[
y = 19.730588892721208 \sqrt{x} - 7.116197572583441
\][/tex]
So, in the blank spaces of your problem:
[tex]\[
y = 19.730588892721208 \sqrt{x} - 7.116197572583441
\][/tex]
