Boat A leaves a dock headed due east at 1:00PM traveling at a speed of 13 mi/hr. At 2:00PM, Boat B leaves the same dock traveling due north at a speed of 20 mi/hr. Find an equation that represents the distance d in miles between the boats and any time t in hours for t≥1, using that t=0 corresponds to the time that Boat A leaves the dock.

To find the equation that represents the distance between Boat A and Boat B at any time t, we can use the concept of relative velocity.

Let's consider the distance traveled by Boat A in t hours. Since Boat A is traveling at a constant speed of 13 mi/hr due east, the distance traveled by Boat A in t hours is given by: dA = 13t (equation 1).

Now, let's consider the distance traveled by Boat B in (t-1) hours since it leaves the dock at 2:00 PM, which is 1 hour after Boat A. Since Boat B is traveling at a constant speed of 20 mi/hr due north, the distance traveled by Boat B in (t-1) hours is given by: dB = 20(t-1) (equation 2).

To find the distance d between the boats, we can use the Pythagorean theorem, which states that the square of the hypotenuse (d) is equal to the sum of the squares of the other two sides (dA and dB). Therefore, we have:

d^2 = dA^2 + dB^2
d^2 = (13t)^2 + (20(t-1))^2
d^2 = 169t^2 + 400(t-1)^2
d^2 = 169t^2 + 400(t^2 - 2t + 1)
d^2 = 169t^2 + 400t^2 - 800t + 400
d^2 = 569t^2 - 800t + 400 (equation 3)

Thus, the equation that represents the distance d in miles between Boat A and Boat B at any time t ≥ 1 is given by equation 3: d^2 = 569t^2 - 800t + 400.

To find an equation that represents the distance between the boats, we can use the concept of relative motion. Here's how to approach it:

1. Determine the positions of Boat A and Boat B at any given time. Since Boat A travels east at a constant speed of 13 mi/hr, its position can be represented as (13t, 0) after t hours, where the x-coordinate represents the distance traveled east and the y-coordinate represents the distance traveled north.

2. Similarly, Boat B travels north at a constant speed of 20 mi/hr, so its position can be represented as (0, 20(t-1)) after t-1 hours. The (t-1) accounts for the fact that Boat B leaves the dock one hour later than Boat A.

3. To find the distance between the boats, we can use the distance formula. The distance (d) between two points (x1, y1) and (x2, y2) in a 2D plane can be calculated using the formula: d = sqrt((x2-x1)^2 + (y2-y1)^2).

4. Applying the distance formula to the positions of Boat A and Boat B, we get: d = sqrt((13t - 0)^2 + (0 - 20(t-1))^2) = sqrt(13^2t^2 + 20^2(t-1)^2).

Therefore, the equation that represents the distance (d) in miles between the boats at any time (t) in hours is:
d = sqrt(13^2t^2 + 20^2(t-1)^2).

Note: This equation assumes that the boats are traveling in a straight line and that there are no external factors, such as wind or current, affecting their motion.

Boat A east:

x = 13 t
Boat B north
y = 20 (t-1)

d = sqrt (x^2+y^2)

d = sqrt [169 t^2 + 400 (t^2-2 t +1) ]

d = (569 t^2 -800 t + 400 )^(1/2