oil is pumped continuously from a well at a rate proportional to the amount of oil left in the w

Step 1: Write the differential equation representing the rate at which oil is pumped from the we

Oil is pumped continuously from a well at a rate...

Transcript

00:01 In this situation given data is that is saying that oil is pumped continuously from a well at a rate proportional to proportional to amount of oil that is left in the well and now further saying that initially there were 2 million wells barrels of oil is present in the well and then six year later the oil that is remain left so that is given that is six year later one million barrels that is remained now in the question further you can see that there are two parts given subparts so i am asking both subparts one by one so let's start answering this question so in the sub part a it is asking that that is at what rate was was the amount of oil in the well decreasing when there were 1 million 200 ,000 remaining so now in the question now for subpart a first step is that is first i have to find a equation that is representing the amount of amount of oil that is left after t time t year time so for finding this we have to use that is data given so in the beginning there is a condition given so that is rate is proportional to amount that is left in the well so here i will say that that is rate that is db by d t that is proportional to amount of oil that is left after tea time so that means db by d t will present the rate and be in term of t that will be present the amount of oil that is left in the oil left in the well after tea year and now in the question saying that this rate is proportional to that is amount of oil left in the well so here i have to take a proportional sign now next step is remove this proportional and it will be witness that is tb by d t that is equal to k times of b in term of t k will be here that is proportional constant so now simplify this differential equation so it will be written as in the other way that is t b in term of t that is equal to k times of d t now take an integration both sides so it will return as that is i will say that integral t b in term of t that is equal to integral kd t now simplify it then it will be written as that is alan b in term of t it will be equal to k t plus c here c will be constant so now we have to find the value of c so we have to use the data that is given so in the same that is initially the amount the amount of oil that is left in the well that is you and that is 2 million so that means t that is equal to 0 b at we will be that is given 2 million so put this data in this above step and i will get that is value of c that is so before putting first remove this log so that we will solve the calculation easily so after removing the allen from left side then i will get a result that is b term of t that is equal to e to the power of k t plus c and now put the value of t put this data that is t that is equal to zero and then b will be that is equal to two million then i will get that is e to the power of c that is equal to two million so i will say that this is a equation number one now next step is that is now i am writing the another data that is given so saying that after six year later the amount of file that is left that is given that is one million so here i will say that t that is equal to zero sorry t that is equal to six then i will get then b6 that will be that is one million in equation of be in term of t so that means in this equation so it will be written as that is be in term of that is one million that will be equal to e times of e to the power of sorry e to the power of k multiplied by six and then plus c so e to the power of k plus k t plus c that will be written as in the form that is e to the power of k t multiplied by e to the power of c so here i had that is after putting the value of t that is equal to 6 then it will be written as as that is e to the power of k times of 6 and now e to the power of c that will be written as from this equation 1 that is put this value here so that is 2 million so now simplify it then i will get the value of k that is proportional constant sorry after simplifying this then i will get the value of 8 to the power of 6 k so that is equal to 1 over 2 this step will come after dividing by 2 million to both sides and changing the both side then i will get e to the power of 6k that is equal to half now next step is i'm writing here the equation v that means amount of oil that is left after t here that is equal to write this equation again here so that means that is e to the power of k t plus c c where i can say that to be written as that is e to the power of k t times of e to the power of c now this is equation one and assume that this is a equation two so from equation one and equation two it will be written as that is that is b in term of t that is equal to half to the power of t then e to the power of c that will be written as that is multiplied by 2 million so what i can say that amount of oil left after t year that is equal to that is here you can see that this is the value of e to the power of 6k and now here i had taken a slight mistake please correct this that is here i have have to put that is here you consider from equation to the value is given that is value which i had found that is half that will be equal to e to the power of 6k 6k so now it will be tiness that is e to the power of k that will be written as that will be turned as half to the power of 1 over 6 so now e to the power of k t that will be that is half to the power of t over 6 and multiplied by 2 million so what i can say that it is written as amount of oil that is left after t here that is equal to 2 million times of half to the power of t over 6 so this is the equation that is giving the amount of oil that is left out of after the year.

11:04 Now in the question is i seen that in part a that is we have to find the rate when the oil remaining that is given 1 million 200 ,000 so for finding the rate we have to take a first derivative of b in term of t about t so that means b that's t that will be equal to that is two thousand two thousand two sorry 2 million multiplied by differentiation of half to the power of t over 6 about t so we know that the property of formula for differentiation that is here i'm adding it is differentiation formula that is d by d x of a to the power of x that is equal to again i am writing here so it will be written as this step that is in more easy way so that means b -t that is equal to 2 million times of d by d t of half to the power of 1 over 6 multiplied by t now here assume that a will be that is after comparing with this standard formula differentiation formula so a will be that is after the power of 1 over 6 and x will be that is equal to t so after taking the differentiation about t then i will get in the result that is b dash t that is equal to 2 million multiplied by a to the power of x that means half to the power of 1 over 6 times of t or i can say that t over 6 then alan of here basis that is e so allen of a that means half with the power of 1 over 6 or i can say that it will be written as that is b test t that will be equal to 2 million multiplied by 2 to the power of negative t over 6 and this will be written as that is multiplied by we know that the allen property that is lenn of a to the power of b that will be equal to b times of a so here use this property then it will return that is multiplied by 1 over 6 times of allen of half or i can say that it will be written in the other way that is let is use this equation so here i will say that that is it will be witness that is b that is equal to this will be written as that is b in term of t so multiplied by 1 over 6 times of allen of half so now next step is that is in the question in part a that is given we have to find the rate at which the amount of oil that is left that is given that is 1 ,000 sorry 1 million 200 ,000 so here put this value of b in term of t that is equal to that is equal to 1 million 200 thousand multiplied by 1 over 6 multiplied by ln of half so simplify it then i will get b dash t or rate so that is equal to negative of 1 million 386 sorry that is negative of 138 ,000 629 .4 so here the sign is showing that that is decreasing so you can see that the rate is decreasing that is value will be that is 138 629 .44 so this is the answer for part a now that means an unit will be that is barrels per year now come to the next part so in the part b it is asking that when will be when will there be 1 100 ,000 barrels remaining now i will give here answer for part b so for answering for it it is asking that we have to find the time after it the amount of oil that will be remained that is one million for the 100 ,000 so let me see put the value of b in term of t that is equal to so in that amount of oil that is left after t l so put that is equal to 100 ,000 so then you write the equation again so that is b in term of t that is equal to some previous step that is be in term of t that is equal to 2 million multiplied by half to the power of t over 6 so here i will write it is 2 million multiplied by half to the power of t over 6 and substitute here the value of b in term of t that is equal to 100 ,000 so that is equal to 200 million multiplied by half to the power of t over 6...

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