Two ships leave the same port at noon. Ship A sails north at 12 mph, and ship B sails east at 19 mph. How fast is the distance between them changing at 1 p.m.? (Round your answer to one decimal place.)
d^2=(12t)^2 + (19t)^2 where t is time in hours sailing. In this case, t=1hr
solve for d.
d^2=(12t)^2 + (19t)^2 where t is time in hours sailing. In this case, t=1hr
take the derivative
2d dd/dt=2*144t+2*19^2 t
dd/dt = relative velocity
v= (288 +261)/d
where d=sqrt(12^2+19^2)
To determine how fast the distance between the two ships is changing at 1 p.m., we can use the Pythagorean theorem and the concept of rate of change.
Let's consider the position of the ships at 1 p.m. Ship A will have traveled for 1 hour at a speed of 12 mph, so it will be 12 miles north of the starting point. Ship B will also have traveled for 1 hour at a speed of 19 mph, so it will be 19 miles east of the starting point.
Now, we can visualize the situation on a coordinate plane. Let's assume the starting point is the origin (0,0). Ship A will be located at (0, 12), and Ship B at (19, 0).
To find the distance between the two ships, we can use the distance formula:
Distance = square root of ((x2 - x1)^2 + (y2 - y1)^2)
In this case, x1 = 0, y1 = 12, x2 = 19, and y2 = 0.
Distance = square root of ((19 - 0)^2 + (0 - 12)^2)
Distance = square root of (361 + 144)
Distance = square root of 505
Distance ≈ 22.47 miles
Now, let's differentiate the distance formula with respect to time (t) to find how fast the distance is changing:
d(Distance)/dt = (d/dt) [ square root of ((x2 - x1)^2 + (y2 - y1)^2) ]
Since x1 = 0, y1 = 12, x2 = 19, and y2 = 0, we can substitute these values into the distance formula:
d(Distance)/dt = (d/dt) [ square root of ((19 - 0)^2 + (0 - 12)^2) ]
d(Distance)/dt = (d/dt) [ square root of (361 + 144) ]
d(Distance)/dt = (d/dt) [ square root of 505 ]
To differentiate the square root function, we can use the chain rule. Let's denote the square root of 505 as z, so the equation becomes:
d(Distance)/dt = (d/dz) (square root of z) * (dz/dt)
The derivative of the square root of z with respect to z is 1 / (2 * square root of z).
So, the equation becomes:
d(Distance)/dt = (1 / (2 * square root of z)) * (dz/dt)
Now, we need to find dz/dt. From the given information, we know that Ship A is sailing north at a constant speed of 12 mph, and Ship B is sailing east at a constant speed of 19 mph. Since both ships are continuously moving away from the starting point, the x-coordinate of Ship A is increasing at a rate of 0 mph, and the y-coordinate of Ship B is increasing at a rate of 0 mph. Therefore, dz/dt is 0.
Substituting this value into the equation, we have:
d(Distance)/dt = (1 / (2 * square root of z)) * 0
d(Distance)/dt = 0
Therefore, the distance between the two ships is not changing at 1 p.m. The rate of change is 0 mph.
To find how fast the distance between the two ships is changing, we can use the concept of rate of change.
Let's assume that the position of Ship A at time t is given by the function A(t) in miles, and the position of Ship B at time t is given by the function B(t) in miles.
Since Ship A sails north at 12 mph, the rate of change of its position can be represented as A'(t) = 12 mph. Similarly, since Ship B sails east at 19 mph, the rate of change of its position can be represented as B'(t) = 19 mph.
The distance between the two ships at time t can be found using the distance formula in two dimensions:
D(t) = √((A(t) - B(t))^2 + (A(t) - B(t))^2)
To find how fast the distance is changing, we need to find the derivative of the distance function D(t) with respect to time t.
dD/dt = d/dt √((A(t) - B(t))^2 + (A(t) - B(t))^2)
To simplify the derivative, we can square the equation inside the square root:
dD/dt = d/dt ((A(t) - B(t))^2 + (A(t) - B(t))^2)^1/2
Using the chain rule, the derivative can be calculated as:
dD/dt = 1/2 ((A(t) - B(t))^2 + (A(t) - B(t))^2)^(-1/2) * (2 * (A'(t) - B'(t))(A(t) - B(t)) + 2 * (A(t) - B(t))(A'(t) - B'(t)))
Substituting the known values:
A'(t) = 12 mph
B'(t) = 19 mph
We can now evaluate the derivative at t = 1 hour (since the ships leave at noon and we need to find the rate at 1 p.m.):
dD/dt = 1/2 ((A(1) - B(1))^2 + (A(1) - B(1))^2)^(-1/2) * (2 * (12)(A(1) - B(1)) + 2 * (A(1) - B(1))(19))
To find the values of A(1) and B(1), we need to consider the distance traveled by each ship in one hour.
Ship A travels at a constant speed of 12 mph for 1 hour, so A(1) = 12 miles.
Ship B travels at a constant speed of 19 mph for 1 hour, so B(1) = 19 miles.
Now we can substitute the values:
dD/dt = 1/2 ((12 - 19)^2 + (12 - 19)^2)^(-1/2) * (2 * (12)(12 - 19) + 2 * (12 - 19)(19))
Simplifying further:
dD/dt = 1/2 ((-7)^2 + (-7)^2)^(-1/2) * (2 * (12)(-7) + 2 * (-7)(19))
dD/dt = 1/2 (98 + 98)^(-1/2) * (2 * (-84) + 2 * (-133))
dD/dt = 1/2 (196)^(-1/2) * (-168 - 266)
dD/dt = 1/2 (196)^(-1/2) * (-434)
Now we can calculate the final answer:
dD/dt = -434/2 * (196)^(-1/2)
Using a calculator, we find:
dD/dt ≈ -13.94 mph
Therefore, the distance between the two ships is changing at a rate of approximately -13.94 mph at 1 p.m.