A ship is 400 miles directly north of the tahiti and is sailing south at 20mi/hr. another ship is 300 miles east of tahiti and is sailing west at 15mi/hr. at what rate is the distance between the ships changing?
this teacher likes negative derivatives
h^2 = x^2 + y^2
2 h dh/dt = 2 x dx/dt + 2 y dy/dt
dx/dt = -15
dy/dt = -20
h at t = 0 is 500
2 (500)dh/dt = 2 (300)(-15) + 2(400)(-20)
enough
To find the rate at which the distance between the ships is changing, we can use the derivative of the distance formula. Let's assume the distance between the ships at a given time t is represented by the variable D.
We can use the Pythagorean theorem to find the distance between the ships:
D^2 = (400 - 20t)^2 + (300 - 15t)^2
Expanding and simplifying this equation, we get:
D^2 = 400^2 - 16000t + 400t^2 + 300^2 - 9000t + 225t^2
Combining like terms, we have:
D^2 = 625t^2 - 25000t + 250000
To find the rate at which the distance between the ships is changing, we can take the derivative of D^2 with respect to time t:
2D * dD/dt = 1250t - 25000
Rearranging the equation to solve for dD/dt:
dD/dt = (1250t - 25000) / (2D)
To find the rate at which the distance between the ships is changing when t = 0, we substitute t = 0 into the equation:
dD/dt = (1250(0) - 25000) / (2D)
Simplifying, we get:
dD/dt = -25000 / (2D)
Now we need to find the value of D. Using the Pythagorean theorem, we can calculate the initial distance between the ships when t = 0:
D^2 = (400 - 20(0))^2 + (300 - 15(0))^2
D^2 = 400^2 + 300^2
D^2 = 250000
D = 500
Substituting D = 500 into the equation for dD/dt, we get:
dD/dt = -25000 / (2 * 500)
dD/dt = -50 mi/hr
Therefore, the distance between the ships is changing at a rate of -50 miles per hour. The negative sign indicates that the ships are getting closer to each other.
To find the rate at which the distance between the ships is changing, we can use the concept of the Relative Velocity.
The first step is to determine the velocity of each ship with respect to the other ship.
Let's consider the ship that is initially 400 miles north of Tahiti. Since it is sailing south at 20 miles per hour, its velocity relative to the other ship is 20 mi/hr south.
Similarly, for the ship that is initially 300 miles east of Tahiti and sailing west at 15 miles per hour, its velocity relative to the other ship is 15 mi/hr west.
Now, we can use the Pythagorean theorem to determine the distance between the ships. Let's call this distance "d". The distance between the ships can be calculated using the formula:
d^2 = (400 + 20t)^2 + (300 - 15t)^2
where "t" represents the time in hours.
To find the rate at which the distance between the ships is changing, we need to differentiate the equation above with respect to time "t". But before doing that, let's simplify the equation:
d^2 = 16,000 + 1,600t + 400t^2 + 9,000 - 900t + 225t^2
Combining like terms:
d^2 = 16,000 - 500t + 625t^2
Now, differentiate both sides of the equation with respect to time "t":
2d(dd/dt) = -500 + 1,250t
Simplifying:
2(dd/dt) = (-500 + 1,250t) / d
Finally, to find the rate at which the distance between the ships is changing, we need to substitute the value of "d" (distance between the ships) and find the rate at a specific time "t".