Question

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.

Answer 1

To find the location and distance between the oil tanker and luxury liner at a given time, use equations of motion and the Pythagorean theorem.

Step-by-step explanation:

To find the location of the oil tanker and luxury liner, we can use the equations of motion. Let's assume that the oil tanker and luxury liner have traveled t hours after 2 p.m. In t hours, the oil tanker will have traveled 14t kilometers west from the origin, and the luxury liner will have traveled 16t kilometers north from the origin. To find the distance d between the oil tanker and the luxury liner at that time, we can use the Pythagorean theorem. The distance is given by d = sqrt((14t)^2 + (16t)^2).

Answer 2

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)