Related Rates Problem

[kjbohn]

Suppose two students start from the same point and begin walking. One student walks directly north at 3m/s and the second students walks slightly southeast(angle is pi/6 below the x-axis) at 4m/s. How fast is the distance between them changing after 10 seconds?Suppose the student walking north gradually speeds up by walking 3+(1+(t-10)^2)^(-1))m/s. How fast is the distance between them changing after 10 seconds?

-I drew a triangle that has the angle inbetween them being 120 degrees or 2pi/3 radians.And I figured out what the distance between the two is so far through law of cosines. I am at a loss of what to do next. Please help.

 

[galactus]

If you're thinking law of cosines, then you're on the right path.

How fast is the distance between them changing after 10 seconds

\(\displaystyle c^{2}=a^{2}+b^{2}-2abcos(\frac{2\pi}{3})\)

Differentiate implicitly:

\(\displaystyle 2c\frac{dc}{dt}=2a\frac{da}{dt}+2b\frac{db}{dt}+\frac{1}{2}\left(a\frac{db}{dt}+b\frac{da}{dt}\right)\).....[1]

c can be found by using the law of cosines.

The northbound person walks \(\displaystyle 30 \;\ m\) in 10 sec. This is side a.

The southeast bound person walks \(\displaystyle 40 \;\ m\) in 10 sec. This is side b.

\(\displaystyle c=\sqrt{30^{2}+40^{2}-2(30)(40)(-1/2)}=10\sqrt{37} \;\ m\)

Enter your knowns into [1] and solve for dc/dt.

Let's see how you proceed on the other part of the problem. Okey-doke.

 

[kjbohn]

for a=3+(1+(t-10)^2)^(-1)m/s would you plug that in to the implicit diffrentiation formula? And when I plug in my knowns for 2c(dc/dt)=2a(da/dt)+2b(db/dt)+1/2(a(da/dt)+(b(db/dt)) what do I do with da/dt and db/dt? Thank you for your help!

 

[Deleted member 4993]

for a=3+(1+(t-10)^2)^(-1)m/s would you plug that in to the implicit diffrentiation formula? And when I plug in my knowns for 2c(dc/dt)=2a(da/dt)+2b(db/dt)+1/2(a(da/dt)+(b(db/dt)) what do I do with da/dt and db/dt? Thank you for your help!

What are "a" & "b" - i.e. what are their physical interpretation?

Are those changing? If yes - at what rate?