00:01
Okay, so in this question, it is given that a dimension of the rectangular metal seat is 6 meter by 3 meter, right? so here, shorter side is 3 meter and the longer side is 6 meter.
00:20
Right, and we are making an open box from this metal seat by cutting square from each corner, right? that we can see from the figure.
00:29
And here the length of each square corner is x right and we need to find the value of x that will give the maximum volume of the box right so from the figure you can see that here length of the box will be 6 minus 2x and width of the box will be 3 minus 2x right and height of the box is x so we can write here dimension that is l is equal to 6 minus 2x, w is equal to 3 minus 2x, and the h is equal to x.
01:10
Okay, now the volume of the box is given by length, multiply by width, multiply by height.
01:19
Now substitute all these three values over here.
01:27
Now simplify this equation.
01:30
So here are 6 multiply by 3, that is 18, 3 multiply by 2x that is 6x, 6 multiplied by 2x is 12x, 6x, 6 multiplied by 2x is 12x, and 2x multiply by 2x that is 4x square minus 6x minus 12x that is equal to minus 18 x now simplify this right okay now we need to find the value of x that will give the maximum volume and to find that we need to first find at the extreme point of the volume function and to find the extreme point we need to take a derivative is equal to 0 take the derivative of the volume function.
02:39
Here 4 is constant, derivative of x cubic is 3x square, minus 18 is constant, derivative of x square will be 2x, and derivative of 18 will be 0.
02:50
Now simplify this...