The three factors that can be varied on the catapult are the start angle, stop angle, and pivot height.

Each factor has three possible levels. A full 33 factorial would allow estima-

tion of all coecients in the general quadratic model, but it would require 27 experiments with no replicates to check the model lack-of-t.

(a) Construct a face-centered cube (central composite) design for this prob-

lem.

(b) Construct a Box-Behnken design for this problem.

(c) Since there are only 27 candidate design points, construct a 10-run design that is a D-optimal subset of the 27 candidates. Suggest one of the points that could be replicated three times to test lack of t.

(d) Use the Vdgraph package to make variance dispersion graphs or fraction of design space plots for each design. What design do you prefer? Why?

(e) Use a catapult to actually run the experiments you have planned.

(f) Fit the general quadratic model to the data, using the rsm function and check the adequacy of the model.
(g) Using numerical optimization, contour plots, or simply evaluating your prediction equation over the 27 possible settings, determine the settings (one of the 27 possible) that you predict would result in a distance closest to exactly 12 feet.
(h) Test the settings you determined in (g). Is the distance close to your predictions?