At 7:00 am, one ship was 60 miles due east from a second ship.If the first ship sailed west at 20 mph, and the second ship sailed southeast at 30 mph, at what time are they closest together?

how did you derive where B is at where did the answer 21.2 come from?

To find the time at which the two ships are closest together, we need to determine when the distance between them is minimized.

We can represent the distance between the two ships as a function of time. Let's consider the first ship as Ship A and the second ship as Ship B.

At any given time t, the distance between the two ships can be determined using the Pythagorean theorem:

Distance(t) = √((x_A - x_B)^2 + (y_A - y_B)^2)

Given that Ship A is initially 60 miles due east from Ship B, Ship A will have sailed 20t miles west, and Ship B will have sailed 30t miles southeast.

Now let's substitute these values into our distance equation:

Distance(t) = √((60 - 20t)^2 + (0 - 30t)^2)

To find the time at which the distance between the ships is minimized, we can differentiate the distance function with respect to t and set it equal to zero:

d(Distance(t))/dt = 0

Let's differentiate the equation:

d(Distance(t))/dt = (-2(60 - 20t)(20) - 2(0 - 30t)(30))/2√((60 - 20t)^2 + (0 - 30t)^2)

Simplifying, we get:

(-1200 + 40t + 1800t)/√((60 - 20t)^2 + (0 - 30t)^2) = 0

Combine like terms:

(-1200 + 1840t)/√((60 - 20t)^2 + (0 - 30t)^2) = 0

To solve for t, we can multiply both sides of the equation by the denominator:

-1200 + 1840t = 0

Rearranging the equation, we have:

1840t = 1200

t = 1200 / 1840

Simplifying, we find:

t ≈ 0.6522 hours

To convert this to minutes, we multiply by 60:

t ≈ 0.6522 * 60

t ≈ 39.13 minutes

Therefore, the two ships will be closest together at approximately 39.13 minutes after they start sailing. To determine the exact time, we need to know the initial starting time at 7:00 am.

To find the time when the two ships are closest together, we can use the concept of relative motion. We need to calculate the time it takes for the ships to close the initial distance between them and then determine the time it takes for the ships to reach their closest point.

Let's start by calculating the time it takes for the first ship to close the initial distance of 60 miles. The first ship is sailing west at a speed of 20 mph, so it will take:

Time1 = Distance / Speed = 60 miles / 20 mph = 3 hours

Now, let's calculate the position of the second ship after 3 hours. The second ship is sailing southeast, which means it is moving both south and east. We need to find the distances traveled in each direction.

The second ship is sailing at a speed of 30 mph. However, to calculate the distance traveled east, we need to consider its velocity component in that direction. Suppose its eastward velocity component is v. Then, the eastward distance traveled is:

Distance east = v * Time1

Since the second ship is also moving southward, we need to consider the vertical components as well. The southward distance traveled is given by:

Distance south = Speed * Time1 = 30 mph * 3 hours = 90 miles

Now, we need to find the time it takes for the second ship to travel the combined distance east and south until it reaches the initial position of the first ship. To calculate this, we can use the Pythagorean theorem:

Distance east^2 + Distance south^2 = Total distance^2

(v * Time2)^2 + (90 miles)^2 = 60 miles^2

Simplifying the equation, we have:

v^2 * Time2^2 + 8100 miles^2 = 3600 miles^2

v^2 * Time2^2 = 3600 miles^2 - 8100 miles^2

v^2 * Time2^2 = -4500 miles^2

Since distance cannot be negative, we conclude that the second ship cannot reach the initial position of the first ship. Therefore, the ships do not get any closer together after the initial 3 hours.

Hence, the two ships are closest together at the time of 7:00 am + 3 hours = 10:00 am.

Assume the ships started at A=(60,0) B=(0,0)

at time x hours,

A is at 60-20x
B is at (21.2x,-21.2x)

the distance d is thus given by

d^2 = (60-20x-21.2x)^2 + (-21.2x)^2

2d dd/dx = 2(60-41.2x)(-41.2) + 2(-21.2x)(-21.3)

we can forget about the denominator and just set the numerator = 0, to get dd/dx = 0 when x = 1.151, = 1:09

So, the ships are closest at 8:09 am