 Hello and welcome to the session. In this session we discuss the following question which says a tank with rectangular base and rectangular sides open at the top is to be constructed so that its depth is 2 meters and volume is 8 meter cube. If building of tank costs rupees 70 per square meter for the base and rupees 45 per square meter for sides what is the cost of least expensive tank? Before we move on to the solution let's recall some formulae. First we have area of a rectangle is equal to length into breadth then area of the four walls is equal to 2 into the height h this whole into length plus breadth. Then we have volume of cuboid is equal to length into breadth into height. This is the key idea that we use for this question. Let's move on to the solution now. So this is the rectangular tank which is to be constructed. It has rectangular base and rectangular sides and it is open at the top. We have to find the cost of least expensive tank. We assume let the length of the base of the rectangular tank be equal to x meters. That is this is x meters and we assume let the breadth of the base of the rectangular tank be equal to y meters. So this means this is also y. It is given that the depth of the tank is 2 meters. So this is 2 and we take let s be the total expense of building the tank. Now we are given that the volume of the tank is equal to 8 meter cube. Let s first find out the area of the base of the rectangular tank which would be equal to its length into its breadth. That is x into y that is x y meter square is the area of the base of the rectangular tank. Now this rectangular tank has four sides which are rectangular. So now we have area of the four sides of the rectangular tank is equal to 2 into the depth of the rectangular tank. That is 2 into length of the rectangular tank x plus the breadth of the rectangular tank y. So this is equal to 4 into x plus y the whole meter square is the area of the four sides of the rectangular tank. It is given in the question that for the base the cost of building the tank is rupees 70 per square meter. Now the area of the base of the rectangular tank is x y meter square. So for the base of area x y meter square the cost of building the tank is equal to rupees 70 into x y. So that is rupees 70 x y is the cost of building the tank with base area of x y meter square. Then we have for the sides the cost of building the tank is given to be rupees 45 per square meter. Now the area of the four sides of the rectangular tank is 4 into x plus y the whole meter square. So for the four sides of area 4 into x plus y the whole meter square the cost of building the tank is equal to rupees 4 into x plus y the whole into 45. And this would be equal to rupees 180 into x plus y the whole. We had assumed earlier that S be the total expense of building the tank. So now we have this S would be equal to the cost of building the tank for the base of area x y meter square which is 70 x y plus the cost of building the tank for the four sides of area 4 into x plus y meter square which is 180 into x plus y. So we have S is equal to 70 x y plus 180 into x plus y the whole. Now we know that the volume of the tank is equal to 8 meters cube and volume of the tank is L into B into H that is length into bit into height is equal to 8. Now length into bit is the area of the base which would be x into y into H so this is equal to 8. H is 2 that is 2 meters so from here we get x into y is equal to 8 upon 2 that is 4. Therefore x y is equal to 4 and from here we get y is equal to 4 upon x. Now substituting value of y in S we get S is equal to 70 x into y that is 4 upon x plus 180 into x plus y that is 4 upon x. Now this x cancels with this x so S would be equal to 280 plus 180 into x plus 4 upon x the whole. Since we need to find the cost of least expensive tank so first we will differentiate this S with respect to x. So differentiating both sides with respect to x we get dS upon dx is equal to 0 plus 180 into 1 minus 4 upon x square. Thus we have dS upon dx is equal to 180 into 1 minus 4 upon x square. Now for least expenses that we need to find out we will put dS upon dx equal to 0. This means 180 into 1 minus 4 upon x square the whole would be equal to 0. Thus 1 minus 4 upon x square is equal to 0 or further we have 4 upon x square is equal to 1 this gives us x square is equal to 4. And so now we have x is equal to 2 we won't be taking negative 2 because it is the length of the rectangle base and length cannot be negative so x is equal to 2. Now next we find d2S by dx2 this we find out by differentiating this with respect to x again and so this is equal to 180 into 8 upon x cube. Thus we get d2S by dx2 is equal to 1 4 4 0 upon x cube. Now d2S by dx2 at x equal to 2 is equal to 1 4 4 0 upon 2 cube that is 8 and 180 8 times is 1 4 4 0 and this is positive that is greater than 0. Now since d2S by dx2 at x equal to 2 is greater than 0 therefore s is least at x equal to 2. s is least means that the cost of or say the total cost of building the tank is least at x equal to 2. Now since we need to find the least cost this would be equal to the value of s at x equal to 2. So in this equation we will put x equal to 2 so least cost would be equal to 280 plus 180 into x that is 2 plus 4 upon x that is 4 upon 2. Now this 2 2 times is 4 and this is equal to 280 plus 180 into 4 this is equal to 280 plus 720 and this is equal to 1000. So the least cost is equal to rupees 1000 or you can say that the cost of least expensive tank is equal to rupees 1000. So this is our final answer with this we complete the session hope you have understood the solution of this question.