Two parallel paths 26 m apart run east-west through the woods. Brooke jogs east on one path at 2 km/h, while Jamail walks west on the otherpath at 3
km/h. If they pass each other at time 𝑡=0, how far apart are they 5s later, and how fast is the distance between them changing at that moment?
The distance between them at 𝑡=5
is 𝐿
I converted from km/h to m/s 2km/h = 0.555m/s and 3km/h = 0.833 m/s and I'm not sure where to go from there.
At time t, the distance z is
z^2 = (.555t+.833t)^2 + 26^2 = 1.927t^2 = 26^2
z dz/dt = 3.853t
now just find z(5) and solve for dz/dt
To find the distance between Brooke and Jamail 5 seconds later, we can calculate the distance each of them traveled in 5 seconds.
Brooke is jogging east at a speed of 0.555 m/s. In 5 seconds, she would have covered a distance of:
DistanceBrooke = SpeedBrooke * Time = 0.555 m/s * 5 s = 2.775 m
Jamail is walking west at a speed of 0.833 m/s. In 5 seconds, he would have covered a distance of:
DistanceJamail = SpeedJamail * Time = 0.833 m/s * 5 s = 4.165 m
To determine how far apart they are after 5 seconds, we need to subtract the distance each of them covered from the initial distance of 26 m between the paths:
Distance_after_5_seconds = 26 m - (DistanceBrooke + DistanceJamail) = 26 m - (2.775 m + 4.165 m) = 26 m - 6.94 m = 19.06 m
Therefore, the distance between Brooke and Jamail after 5 seconds is 19.06 meters.
To find how fast the distance between them is changing at that moment, we need to find the rate of change of the distance between them. We can do this by finding the derivatives of their distances with respect to time.
The rate of change of the distance between Brooke and Jamail after 5 seconds can be calculated as:
Rate_of_change = d(Distance_after_5_seconds)/dt
To find this, we can differentiate the equation for the distance after 5 seconds:
Rate_of_change = d(26 m - (DistanceBrooke + DistanceJamail))/dt
Differentiating with respect to time:
Rate_of_change = d(26 m - (2.775 m + 4.165 m))/dt
Since the initial distance and speeds are constants, the derivatives of these values with respect to time are zero:
Rate_of_change = d(26 m - 6.94 m)/dt = d(19.06 m)/dt = 0
Therefore, the rate of change of the distance between Brooke and Jamail after 5 seconds is zero.
To find the distance between Brooke and Jamail at time 𝑡=5, we need to determine how far each of them has traveled in 5 seconds.
Brooke is jogging east at a speed of 0.555 m/s. In 5 seconds, she would have traveled a distance of:
Distance_Brooke = Speed_Brooke * Time = 0.555 m/s * 5 s = 2.775 m
Similarly, Jamail is walking west at a speed of 0.833 m/s. In 5 seconds, he would have traveled a distance of:
Distance_Jamail = Speed_Jamail * Time = 0.833 m/s * 5 s = 4.165 m
To find the distance between them at 𝑡=5, we need to add their individual distances from each other:
Distance_L = Distance_Brooke + Distance_Jamail = 2.775 m + 4.165 m = 6.94 m
So, the distance between Brooke and Jamail at 𝑡=5 is 6.94 meters.
To find how fast the distance between them is changing at that moment, we need to take the derivative of the distance equation with respect to time.
The rate at which the distance between them is changing is given by the derivative of the function Distance_L with respect to time:
Velocity = d(Distance_L) / d(Time)
Since Brooke is moving east and Jamail is moving west, their velocities have opposite signs. Let's set Brooke's velocity as positive and Jamail's velocity as negative.
Velocity = Velocity_Brooke - Velocity_Jamail
Velocity = Speed_Brooke - Speed_Jamail
Velocity = 0.555 m/s - (-0.833 m/s)
Velocity = 0.555 m/s + 0.833 m/s
Velocity = 1.388 m/s
Therefore, the distance between Brooke and Jamail is changing at a rate of 1.388 m/s at 𝑡=5.