Two boats on opposite banks of a river start moving towards each other. they first pass each other 1400 meters from one bank. they each continue to the opposite bank, immediately turn around and start back to the other bank. when they pass each other a second time, they are 600 meters from the other bank. we assume that each boat travels at a constant speed all along the journey. find the width of the river?

S1 = speed of boat 1

 t1 =  time to do 1400 meters (boat 1)

S1*t1 = 1400

S2  = speed of boat 2

1400 + S2*t1 = X

t2 = time to do X + 600 (boat 2)

S1*t2 = X + 600

S2*t2 = 2X - 600

S1 = 1400/t1

S2 = (X-1400)/t1

T = t2/t1  

1400*T = X + 600

X*T - 1400*T = 2X - 600

X = 3600 meters

 River is 3600 meters wide

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.