Determine a power regression for the data given below.
What diameter would give a length of 380.9 ?
Round the numerical values of the function to three decimal places before you compute your final answer.
Round your final answer to one decimal place.
\begin{tabular}{|c|c||c||c||c||c||c|}
\hline Diameter & 13.6 & 16.7 & 20.4 & 24.3 & 25.5 & 27 \\
\hline \hline Length & 205.7 & 239.6 & 278.1 & 317.2 & 328.9 & 343.6 \\
\hline
\end{tabular}

Step 1 ：The problem is asking for a power regression, which is a type of curve fitting method. Power regression is a type of nonlinear regression analysis that is used to model situations in which the response (dependent) variable is proportional to the independent variable raised to a power. The general form of a power function is \(y = ax^b\). In this case, the diameter is the independent variable (x) and the length is the dependent variable (y).

Step 2 ：To find the power regression, we need to find the values of a and b. This can be done by taking the logarithm of both sides of the equation, which transforms the power function into a linear function. We can then use the method of least squares to find the best fit line for the transformed data, and then transform back to get the power function.

Step 3 ：Once we have the power function, we can use it to find the diameter that would give a length of 380.9.

Step 4 ：Given the diameters \([13.6, 16.7, 20.4, 24.3, 25.5, 27.0]\) and the corresponding lengths \([205.7, 239.6, 278.1, 317.2, 328.9, 343.6]\), we can calculate the diameter that would give a length of 380.9.

Step 5 ：The calculated diameter for a length of 380.9 is approximately 31.017350692752565.

Step 6 ：Rounding to one decimal place, the final answer is \(\boxed{31.0}\).