Chapter 13.6, Problem 66E

Ocean Floor

A team of oceanographers is mapping the ocean floor to assist in the recovery of a sunken ship. Using sonar, they develop the model

$D=250+30x^2+50\sin \frac{\pi y}{2},0\le x\le 2,0\le y\le 2$

where D is the depth in meters, and x and y are the distances in kilometers.

(a) Use a computer algebra system to graph D.

(b) Because the graph in part (a) is showing depth, it is not a map of the ocean floor. How could the model be changed so that the graph of the ocean floor could be obtained?

(c) What is the depth of the ship if it is located at the coordinates $x=1$ and $y=0.5?$

(d) Determine the steepness of the ocean floor in the positive x-direction from the position of the ship.

(e) Determine the steepness of the ocean floor in the positive y-direction from the position of the ship.

(f) Determine the direction of the greatest rate of change of depth from the position of the ship.