A ship traveled at an average rate of 22 miles per hour going east. It then traveled at an average rate of 17 miles per hour heading north. If the ship traveled a total of 212 miles in 11 hours, how many miles were traveled heading east?

The unknown parameter that we have to solve is distance. Since we are given with the average speed and total time, we have to use these two to obtain the distance. The equation for this is

distance = speed * time

First thing to do is equate the distances heading east and north with the total distance of 212 miles:

22 miles/hour (x) + 17 miles/hour (y) = 212, where x is the time it took to head east while y is the time it took to head north. This is our first independent equation. Since we have two unknowns (x and y), we have to formulate one more equation in order for the system to be solvable. The second equation would be the total time:

x + y = 11 hours

Rearranging the equation: y = 11 - x. We substitute this to the first equation so that only one variable is used. Then, we can solve for x.

22x + 17 (11-x) = 212

Solving for x,

x = 5 hours

Therefore, y = 11 - 5 = 6 hours

From here, we can already calculate the total distance covered when heading east. That is equal to 22 miles/hour * 5 hours = 110 miles.