At 3 p. m. an oil tanker traveling west in the ocean at 14 kilometers per hour passes the same spot as a luxury liner that arrived at the same spot at 2 p. m. while traveling north at 16 kilometers per hour. if the "spot" is represented by the origin, find the location of the oil tanker and the location of the luxury liner t hours after 2 p. m. then find the distance d between the oil tanker and the luxury liner at that time.

N (t) = 16t; Distance north of spot at time t for the liner. W (t) = 14 (t-1); Distance west of spot at time t for the tanker. d (t) = sqrt (N (t) ^2 + W (t) ^2); Distance between both ships at time t. Let's create a function to express the distance north of the spot that the luxury liner is at time t. We will use the value t as representing "the number of hours since 2 p. m." Since the liner was there at exactly 2 p. m. and is traveling 16 kph, the function is N (t) = 16t Now let's create the same function for how far west the tanker is from the spot. Since the tanker was there at 3 p. m. (t = 1 by the definition above), the function is slightly more complicated, and is W (t) = 14 (t-1) The distance between the 2 ships is easy. Just use the pythagorean theorem. So d (t) = sqrt (N (t) ^2 + W (t) ^2) If you want the function for d () to be expanded, just substitute the other functions, so d (t) = sqrt ((16t) ^2 + (14 (t-1)) ^2) d (t) = sqrt (256t^2 + (14t-14) ^2) d (t) = sqrt (256t^2 + (196t^2 - 392t + 196)) d (t) = sqrt (452t^2 - 392t + 196)