00:01
Hello student, according to the question here, we have to find the volume of the box.
00:05
So let's suppose here we have a given box and having a length that is 10.
00:11
So here this is a length of 10.
00:15
Now on the side it is cut to the let's suppose x and here this is also x and cut from the corner side.
00:24
So now from here we have to find the dimension of the box.
00:29
So here let's suppose according to the question that side of the square cut from corners vx.
00:51
So now from here we can write the volume of the box it is given by volume of the box it is equal to length multiplied by breadth multiplied by height.
01:05
So now from here we have a length that is 10.
01:07
So here we can write it as volume of the length here with n minus of x is the remaining volume and breadth that is n minus x multiplied by x.
01:18
Now multiplying from here we have that is 100 minus plus of x square minus of 20x and this is multiplied by x.
01:28
Now from here we have the volume it is equal to x multiplied by x inside the bracket we have x cube minus of 20x square plus of 100x.
01:38
Now from here we get the volume of the box in terms of x.
01:44
So differentiating with respect to x we have v dash it is equal to d divided by dx of the function x cube minus 20x square plus 100x.
01:56
Now from here using the differentiation formula as that is d divided by dx x to the power and it is equal to n x to the power n minus 1.
02:07
Now from here we get the v dash by using the formula that is v dash it is equal to here vx square minus of 20x and plus of 100.
02:20
So for maximum value we have to set that is v dash is equal to 0.
02:31
So here 3x square minus 40x plus 100 it is equal to 0.
02:37
So here we can see this is a quadratic equation.
02:40
So on comparing with x square plus bx plus c it is equal to 0.
02:45
So as we do the formula for this that is x is equal to minus b plus or minus root over b square minus 4ac divided by 2a.
02:55
So now from here after putting those value we got the value of x as that is x is equal to 10 divided by 3, 10.
03:03
Now if we put x equal to 10 divided by 3 and 10 in the given function for the second order derivative...