At t=0, ship A is 12 miles due north of ship B. Ship A travels 12 miles/hour due south, while ship B travels 8 miles/hour due east.
a. Write a function for the distance between the two ships.
b. At what time are the two ships closest?
To solve this problem, we can use the concept of vectors to represent the positions and velocities of the two ships. Let's assign a coordinate system with ship A's initial position at the origin (0, 0).
a. The position vectors for the two ships can be written as:
Ship A: rA(t) = (0, -12t)
Ship B: rB(t) = (8t, 0)
The distance between the two ships can be found using the formula for Euclidean distance:
Distance = √[(xB - xA)^2 + (yB - yA)^2]
Plugging in the expressions for rA(t) and rB(t), we get:
Distance = √[(8t - 0)^2 + (0 - (-12t))^2]
= √[(8t)^2 + (12t)^2]
= √[64t^2 + 144t^2]
= √(208t^2)
= 4√13t
Therefore, the function for the distance between the two ships is:
f(t) = 4√13t
b. To find the time when the two ships are closest, we need to find the minimum value of the distance function.
To do this, we can take the derivative of the distance function with respect to t and set it equal to zero:
f'(t) = 4√13
Setting f'(t) = 0, we have:
4√13 = 0
Since this equation has no solution, the distance function does not have a minimum value.
This means that the two ships are constantly getting farther apart from each other and there is no time when they are closest.
a. Let's assume that the distance function is D(t), where t represents time in hours.
Ship A is initially 12 miles due north of ship B. Let's consider this as the vertical distance between the two ships, which is a fixed value. As ship A travels 12 miles/hour due south, its vertical distance from ship B decreases linearly at a rate of 12 miles/hour.
Ship B travels 8 miles/hour due east, so its horizontal distance from ship A increases linearly at a rate of 8 miles/hour. Since these distances are perpendicular to each other, we can use the Pythagorean theorem to find the distance between the two ships.
At time t, the vertical distance is given by 12 - 12t, and the horizontal distance is given by 8t. Applying the Pythagorean theorem, the distance function becomes:
D(t) = √((12 - 12t)^2 + (8t)^2)
b. To find the time when the two ships are closest, we can find the minimum value of the distance function D(t). This can be done by finding the first derivative of D(t) with respect to t, setting it equal to zero, and solving for t.
Let's find the first derivative of D(t) using the chain rule of differentiation:
D'(t) = (-2(12 - 12t)(-12) + 2(8)(8t)) / (√((12 - 12t)^2 + (8t)^2))
Setting D'(t) = 0 and solving for t:
(-2(12 - 12t)(-12) + 2(8)(8t)) / (√((12 - 12t)^2 + (8t)^2)) = 0
Simplifying and solving for t will give us the time when the two ships are closest.