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?

Step-by-step explanation:

lets say in the shown triangle x=165

so speed

and  

Also since distance = speed* time

time = pi minutes = pi/60 seconds

y =

y =

Now using the pythagorean :

derivate the equation so  we get :

m/s

So Rate of change = 25.387 m/s

x = central angle apc = measure of minor arc ac

the two angles x and 30 add to 180, so x+30 = 180 and x = 180-30 = 150. the measure of minor arc ac is 150 degrees.

what we need to do from here is find the circumference and then multiply it by x/360 since there are 360 degrees in a circle. in this case, x = 150, so x/360 = 150/360 = 15/36 = 5/12. the shortest distance along the curve from a to c is 5/12 of the distance around the entire circle (aka circumference)

c = circumference

c = pi*d

c = pi*24

c = 24pi

now multiply this by 5/12 to get

distance along minor arc ac = (5/12)*(circumference)

distance along minor arc ac = (5/12)*(24pi)

distance along minor arc ac = 10pi

the formula you can use is

l = (x/360)*pi*d

with

l = length of minor arc

x = central angle

d = diameter

which essentially is the steps we took though a bit differently.

step-by-step explanation: