A Capybara needs to get from point A to point B around a circular pool (see diagram). If the Capybara swims half as fast as he can run, what angle should he jump in the pool to minimize the total travel time?
I know that to minimize time traveling you need to find the equation for time then derive and set to zero.
So far I have that time swimming = 2t, time running = t, speed = distance/time
I know that distance traveled on land is r*theta, but I have no idea how to find the distance traveled in water.
Right now my equation is Time spent traveling = r*theta*t + (Distance traveled in water)*2t
How can I find distance traveled by water? And is the rest of what I have done right?
I have that because the question says that the Capybara swims half as fast as he can run, so if it takes him 't' seconds to run a certain distance, it will take him twice as long to swim that distance(2t).
Also, how can I use the cosine law if I don't know the angle to the right of theta, do I use 180-theta?
ok I got that they don't travel the same distance, so now I have S_running=2S_swimming
I used cosine law to find that the distance swimming = sqrt(r^2+r^2-2r^2cos(180-theta)) = rsqrt(2-2cos(180-theta)
Then for for total time I have
t=r*theta/2S_swimming + (rsqrt(2-2cos(180-theta)))/S_swimming . Can you please let me know if what I have is correct before I solve for the minimum time?
I got theta = pi/3 when I solve it, but I plugged in theta=0 and theta = pi and I'm getting the minimum time to be when theta = 0(Capybara swims the while way), but I think the right answer is that the Capybara is suppose to run the whole way. Can anyone confirm/deny this?