hello everybody so today i will talk about calculus the topic today is about application of derivative so now i will show you how to answer our main largest spot from point a on a pen of a strength river three kilometers wide and once to reach point b and we have eight kilometers the stream on the opposite bank as quickly as possible he could roll his boat across the river to point c and then run to b or he could wrote directly to b or he could go to some point d between b and c and then run to b if he can wrote six kilometers both hours and run eight kilometers per hour where should he land to reach b as soon as possible so now i will show you how to answer here is about the figure in here we know that this is about three kilometer of the river and without stream in here the question they want to find where is the location of where is the location of the on cp to make sure that when the man tries to go to a to d and b and the time for this one should be about the shortest so now i will show you how to answer now i will about cd equals to s and we know that this is about the length the length should be about the positive number so i would put s in here now let's consider about a c c d and a d we can see that this is about the right triangle so we want to find the the length of id in here we can use about by target t room in this situation we have ac square plus cd square equals to id sweat so ac we have three we put number three in here cd we have s we put s in here and id that's about the following here so this is about number nine and we put the square root for the both side we have nine plus s square equals to id now let's go to db if we have the value cd we can find the value of db that should be about b b equals to c b minus cd cp we have 8 cd we have s and now we just finished about id and db the next one we need to find the time time that is about the distance we divide by velocity so we can create about a function in here so the distance about id that's it on the river and we know that the velocity on the river we can do that is about six kilometers per hour so id we have a distance we divide by velocity in here that's about number six the next one we have db we run on the bank of the river so we have a minus s and the velocity of running that's about eight so we put i in here and now we just finished about the set up about the function now we want to find the location of the to make sure that we can get the smallest time or the short term in here so we need to find first derivative of the function so we will do first derivative of this one and this one so we have one over six i will put in here first the derivative of the square root of nine plus square we will have about number two multiplied by the square root of nine plus s square later that we continue continue do first the refer to on numerator a divided by a we have number one first derivative of number one you got about zero this one negative s over a if we do first derivative of negative s we have number one with the negative so we put about negative one over a first derivative of 9 plus square square in here we have to ask [Applause] so number two divide by number two now we can put about first the revert2 of the function equals to number zero that means we put this one equals to number zero now we need to find the value of x i will put negative one over a go to the right [Applause] this one we put on the left 6 and the square root of 9 plus s where we go to the right [Applause] we divide both sides by number two next we put the square for the both sides this one we have 16 s square equals to number nine multiply by nine plus s square so number 9 multiply by number 9 and ask square so i will put this one go to the left and we have 16 square minus 9 square we have seven s square equals to 81 later that we divide both sides by number seven if we put about a square root for the both side we have s equal to number nine over the square root of number seven and s equals to negative number nine over the square root of number seven but on the condition we put about x equal to number zero and s is bigger than number zero sorry about that that means that there should be about a positive number so we need to reject about this one and we need to accept about this number in here so after we got about s equals to number nine over the square root of number 7 we can put into the function in here so that should be about this one so we put inside the calculator and we got a plus the square root of number seven over 8 so ask in here that's about the location we can get the smaller time travel from a to b so if they want to find the value we just put about this number going here and this one approach sigma about 3.402 kilometers and the time for this one we have our in here that should be about 1 4 3 3 0 7 2 hours and this is the final answer we need to find in here this is the end thank you for watching