Question

Two boats sail away from a buoy making a 90 degree angle. The first boat is 165 meters away and is traveling at 25 m/s. The second boat has been traveling at 35 m/s for pi mINUTES. What is the rate of change in distance between the two boats?

Answer 1

Explanation:

lets say in the shown triangle x=165

so speed
`(dx)/(dt) =25`

and
`(dy)/(dt) =35`

Also since distance = speed* time

time = pi minutes = pi/60 seconds

y =
`(35\pi \pi )/(60)`

y =
`(7\pi )/(12)`

Now using the pythagorean :

`x^2+y^2=z^2\\so \\165^2+((7\pi )/(12) )^2 =z^2`

derivate the equation so we get :

`x(dx)/(dt) +y(dy)/(dt) =z(dz)/(dt) \\`
`165*25+35*(7\pi )/(12)=\sqrt{165^2+((7\pi )/(12))^2} *(dz)/(dt)`

`(dz)/(dt) =25.387` m/s

So Rate of change = 25.387 m/s