At 2:00PM boat X is heading due west at 15mph towards a dock that is exactly 15 miles away. At the same time boat Y leaves the dock traveling due north at a speed of 20mph. At what time will the boats be closest together?
Let the time passed since the northbound ship left the dock be t hrs
Make a sketch
Distance traveled by northbound boat is 20t
distance from dock of westbound boat is 15-15t.
let the distance between them be D
D^2 = (20t)^2 + (15-15t)^2
= 400t^2 + 225 - 450t + 225t^2
= 625t^2 - 450t + 225
2D dD/dt = 1250t - 450
dD/dt = (625t - 225)/d
= 0 for a max/min of D
625t = 225
t = 225/625 = .36
so the time is .36 hours past 2:00 pm
.36 hrs = 21.6 minutes or 21 minutes 36 seconds
time is 2:21:36 PM
THANK YOU! I get it now!
To find out at what time the boats will be closest together, we need to determine when their distances from each other are minimized. We can start by setting up a coordinate system and tracking the positions of each boat at different times.
Let's assume that the dock is located at the origin (0,0) on the coordinate system. According to the problem, boat X is traveling due west at a speed of 15 mph, which means its position can be described by the equation x = -15t, where x is the distance traveled by boat X and t is the time in hours.
Boat Y, on the other hand, is traveling due north at a speed of 20 mph. Its position can be described by the equation y = 20t, where y is the distance traveled by boat Y.
Now, we need to find the distance between the two boats at any given time. We can use the Pythagorean theorem to calculate the distance between two points in a Cartesian coordinate system:
distance = √((x2 - x1)² + (y2 - y1)²)
For our scenario, the distance between the two boats can be represented as:
distance = √((-15t - 0)² + (0 - 20t)²)
Simplifying this expression further:
distance = √(225t² + 400t²)
= √(625t²)
= 25t
Now we have an equation for the distance between the two boats as a function of time. To find the time when the distance is minimized, we can take the derivative of this equation with respect to time and set it equal to zero.
d(distance)/dt = 0
d(25t)/dt = 0
25 = 0
However, this equation is not possible, which means there is no minimum distance between the two boats. In other words, the boats will never be closest together.
Therefore, the answer is that the boats will never be closest together.