Answer:
The distance between the ghost changes at 10 pm approximately at a rate of 24.981 kilometers per hour.
Step-by-step explanation:
At first we assume that north and east directions both represent positive quantities. Let suppose that [tex]\vec r_{A,o} = (0\,km,0\,km)[/tex] and [tex]\vec r_{B,o} = (5\,km, 0\,km)[/tex]. If both ghosts moves at constant velocity such that [tex]\vec v_{A} = \left(-15\,\frac{km}{h}, 0\,\frac{km}{h} \right)[/tex] and [tex]\vec v_{B} = \left(0\,\frac{km}{h},20\,\frac{km}{h} \right)[/tex], then the final positions of both ghosts are, respectively:
Ghost A
[tex]\vec r_{A} = \vec r_{A,o}+t\cdot \vec v_{A}[/tex] (Eq. 1)
Ghost B
[tex]\vec r_{B} = \vec r_{B,o}+t\cdot \vec v_{B}[/tex] (Eq. 2)
Where [tex]t[/tex] is the time, measured in hours.
Then, the equations of motion of each ghost are, respectively:
Ghost A
[tex]\vec r_{A} = (0\,km,0\,km)+t\cdot \left(-15\,\frac{km}{h}, 0\,\frac{km}{h} \right)[/tex]
[tex]\vec r_{A} = \left(-15\cdot t, 0)\,\,\,\left[km \right][/tex]
Ghost B
[tex]\vec r_{B} = (5\,km, 0\,km)+t\cdot \left(0\,\frac{km}{h}, 20\,\frac{km}{h} \right)[/tex]
[tex]\vec r_{B} = (5, 20\cdot t)\,\,\,\left[km\right][/tex]
Then, the distance between both ghosts is:
[tex]\vec r_{B/A} = (5,20\cdot t)-(-15\cdot t, 0)\,\,\,[km][/tex]
[tex]\vec r_{B/A} =(5+15\cdot t, 20\cdot t)\,\,\,[km][/tex] (Eq. 3)
The magnitude of the relative is represented by the following Pythagorean identity:
[tex]r^{2}_{B/A} = (5+15\cdot t)^{2}+(20\cdot t)^{2}[/tex]
Then, we find the rate of change of the relative distance ([tex]\dot r_{B/A}[/tex]), measured in kilometers per hour, by implicit differentiation:
[tex]2\cdot r_{B/A}\cdot \dot r_{B/A} = 2\cdot (5+15\cdot t)\cdot 15+2\cdot (20\cdot t)\cdot 20[/tex]
[tex]r_{B/A}\cdot \dot r_{B/A} = 15\cdot (5+15\cdot t)+20\cdot (20\cdot t)[/tex]
[tex]\dot r_{B/A} = \frac{75+625\cdot t}{r_{B/A}}[/tex]
[tex]\dot r_{B/A} = \frac{75+625\cdot t}{\sqrt{(5+15\cdot t)^{2}+(20\cdot t)^{2}}}[/tex] (Eq. 4)
If we know that [tex]t = 4\,h[/tex], then the rate of change of the relative distance at 10 PM is:
[tex]\dot r_{B/A} = \frac{75+625\cdot (4)}{\sqrt{[5+15\cdot (4)]^{2}+[20\cdot (4)]^{2}}}[/tex]
[tex]\dot r_{B/A} \approx 24.981\,\frac{km}{h}[/tex]
The distance between the ghost changes at 10 pm approximately at a rate of 24.981 kilometers per hour.
