Answer:
[tex]32.10\ \text{mph}[/tex]
Step-by-step explanation:
Distance traveled by A in 4 hours = [tex]24\times 4=96\ \text{mi}=a[/tex]
Distance traveled by B in 4 hours = [tex]22\times 4=88\ \text{mi}=b[/tex]
Total distance between A and the initial point of B is [tex]40+96=136\ \text{mi}[/tex]
Distance between A and B 4 hours later
[tex]c=\sqrt{136^2+88^2}\ \text{mi}[/tex]
From Pythagoras theorem we have
[tex](a+40)^2+b^2=c^2[/tex]
Differentiating with respect to time we get
[tex]2(a+40)\dfrac{da}{dt}+2b\dfrac{db}{dt}=2c\dfrac{dc}{dt}\\\Rightarrow (a+40)\dfrac{da}{dt}+b\dfrac{db}{dt}=c\dfrac{dc}{dt}\\\Rightarrow \dfrac{dc}{dt}=\dfrac{(a+40)\dfrac{da}{dt}+b\dfrac{db}{dt}}{c}\\\Rightarrow \dfrac{dc}{dt}=\dfrac{(96+40)\times24+88\times 22}{\sqrt{136^2+88^2}}\\\Rightarrow \dfrac{dc}{dt}=32.10\ \text{mph}[/tex]
So, the the distance between the ships at 4 PM is changing at [tex]32.10\ \text{mph}[/tex]
