00:01
The rectangular box say this one and we have to find out the maximum and the volume, minimum of the volume, when the surface area is 1500 square meters and the total edge length is 200 centimeters.
00:25
First of all we have to label three variables so i suppose x is the length of the face facing us and then y is the width and then the z is the how long the box is now using this y and z we can write the volume to the volume is equal to x times y times z now let's first write down an equation for the surface area.
01:08
So how do you find out the surface area? now there are two surfaces involving x and y.
01:21
So that's going to be 2xy is the total surface area of those two surfaces and and then similarly, there are two surfaces involving y and z, so 2 y, z.
01:43
And then similarly, there are two surfaces involving x and z, so 2x, z, and that is the total surfaces, total number of surfaces of the box.
01:57
Now the area of that box that surfaces is 1500.
02:05
So we can divide each side and the other side by two.
02:11
So we're going to divide by two.
02:15
And then we get xy plus y z plus x z equals xz equal 750.
02:23
Now next we also need to come up with an equation to to repression the total length of the edges.
02:33
It's 200 centimeters.
02:36
Now notice that there are four edges of the label x, again four edges of the label y and four edges of the label z.
02:51
So the total length of those 12 edges, can be written as 4x plus 4y plus 4 z equal to 200.
03:08
Now this time we're going to divide the equation by 4 and thus we get x plus y plus z equals to 50.
03:19
Now we're going to solve for y plus z in the second equation.
03:25
So we get y plus z equals to 50 minus x.
03:32
And then using the first equation, we can perhaps considering these two, we can factor out the x.
03:47
So if you factor out the x, we get x times y plus z plus z plus.
03:57
Yz equal to 750.
04:01
So we basically combine these two.
04:08
And then if you do that, we have to factor out x, we can get y plus z and then the last term is y.
04:17
Okay, so now we replace this y plus z, right? y plus z by 50 minus x.
04:30
So then we get x times 50 minus x plus yz equal to 750.
04:43
And then we solve for yz.
04:50
So yz equal to 750 minus x times 50 minus x.
04:59
Now the reason why we did all of that is because we want the objective function.
05:06
This is what we are maximizing and minimizing to be a function of one variable.
05:14
Right now we, this with volume is a function of three variables.
05:20
So we want to reduce that into a function of one variable.
05:25
So now we've found out that yz can be replaced by 750 minus x times 50 minus x.
05:36
So we can rewrite this equals to x times 750 minus x.
05:50
So now the v can be considered as a function of x.
06:00
So the volume as a function of x can be written as 750x minus 50x squared plus x cube.
06:09
Now next we had to find out the domain of this function.
06:13
So what is the domain of this function? perhaps we can get some idea about the domain of the function by considering the second equation that we obtain by writing an equation for the total edge length.
06:33
So we can roughly say that the x has to be between 0 and 50.
06:42
Right let's write it as open interval so that is easier so the domain is between zero and 50...