Boat A leaves a dock headed due east at 1pm travelling at a speed of 10 mi/hr. At 6pm ,boat B leaves the same dock travelling due south at a speed of 25mi/hr.
Find an equation that represents the distance d in miles between the boats and any time t in hours for t ≥ 5 , using that t =0 corresponds to the time that boat A leaves the dock.
thank you
distance = speed * time, so at time t, you have
A: 10t
B: 25(t-5)
so the distance z is
z^2 = (10t)^2 + (25(t-5)^2 = 725t^2 - 6250t + 15625
z = 5√(29t^2 - 250t + 625)
east is +x direction
south is -y direction
x of boat A = 10 (t - 1 ), y of boat 1 remains 0
x of boat B = 0, y = -25 (t - 6)
d = sqrt (x^2 + y^2)
d^2 = 100 (t^2-2t+1) + 625 (t^2 -12 t + 36)
continue
To find an equation that represents the distance between the boats, we can use the concept of relative motion. Let's break down the problem step by step:
1. Boat A starts at 1 pm and travels due east at a speed of 10 mi/hr. This means that by the time Boat B starts at 6 pm, Boat A has been traveling for 5 hours. So, at t = 5 hours, Boat A has covered a distance of (10 mi/hr) * 5 hrs = 50 miles.
2. Boat B starts at 6 pm and travels due south at a speed of 25 mi/hr. To find the distance covered by Boat B at any given time t (t ≥ 5), we need to calculate the distance traveled in the north-south direction since that's the only direction Boat B is moving. We can use the formula: Distance = Speed * Time.
3. Since Boat B starts at 6 pm, we can say t = 0 corresponds to 6 pm. So, for any given time t (t ≥ 5), the time elapsed since Boat B started is (t - 5) hours.
4. Using the formula, the distance covered by Boat B in the north-south direction at time t is (25 mi/hr) * (t - 5) hrs.
5. The distance between the two boats can be found using the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). In this case, the distance between the boats will be the hypotenuse of the triangle formed by the distance traveled by Boat A (50 miles) and the distance traveled by Boat B (in the north-south direction).
6. Using the Pythagorean theorem, the equation becomes: d² = (50 miles)² + [(25 mi/hr) * (t - 5) hrs]².
7. Simplifying the equation gives us: d² = 2500 + 625 * (t - 5)².
Therefore, the equation that represents the distance d in miles between the two boats at any time t (t ≥ 5) is:
d² = 2500 + 625 * (t - 5)².