Two runners at the same point begin running in opposite directions along a circular track of radius $100$m at a speed of $5$m/s. At what rate is the (shortest) distance between them growing after $10$s?
i have done many related rates problems but none like this and have no idea where to even start any help would be appreciated!
Actually I've answered this question before and I cannot comment because I'm new user hers
but Here's the answer
Let's assume they travel with constant velocity $v$, and Radius of circle be $r$
After time $t$ Arc length will be $vt$ $$l=vt=r\theta$$ and angle subtending that arc will be $$\theta=\frac{vt}{r}$$ now shortest distance between them will be $$x=2r\sin\theta$$
Now we have our relation $$x=2r\sin\left(\frac{vt}{r}\right)$$ Now rate of change of shortest distance between them is $$\frac{dx}{dt}=2r\frac{v}{r}\cos\left(\frac{vt}{r}\right)=2v\cos\left(\frac{vt}{r}\right)$$ Now plug in values
Here's a rough diagram if it helps
Note: In figure after time $t$ their respective positions are shown by $A$ and $A'$
$$x=AA'$$ $$r=BA=BA'$$ $$\angle ABO=\angle A'BO=\theta$$
Hint: Find $\frac{d\theta}{dt}$, and $\theta$ at 10 seconds. Then, you can find a equation involving the distance, since you know the radius, and the angle $\theta$ between them, and thus can write a formula for the distance. Take the derivative with respect to $t$, and you should get your answer.
