00:01
Hello, welcome to this lesson in this lesson.
00:03
We have an open rectangular box that is perfect long and it has surface area of 16 square feet.
00:09
So we'll find the dimension of the box that is will find the width and the height such that we have the volume or the gauge volume as large as possible alright, so we have the surface area the formula for surface area which is equals to two times length times the height plus two times the length times the weight plus two times the height and the width alright, so here we have s a that is equals to two l h plus l w plus h w right and given that we have the surface area which is 16 square feet omitting the unit from now we have 16 which is equals to two times the length and the width plus the length times the height.
01:16
This is the length times the weight plus the height times the width and also at the length given that the length is equals to three this is the length is equals to three so we can put them in there okay, so we have three h plus three w plus h w so this is equals to three h plus three w plus h w so here we will make each the subject and you have eight all right, which is equals to in fact we need to have three last last three at this point less send three w to the left side or we subtract three w from both sides we have eight minus three w is close to h this is three plus three plus w.
02:43
Sorry.
02:44
So here let's divide through by three plus w we have three plus two.
02:50
So this causes our parts we have the height which is equals to eight minus three w on three plus two okay, so let this be second equation all right, and we can find the area we've already found the area.
03:12
So let's find the volume which would maximize so the volume is equals to the length times the height times the width and l equals to or at l equals to three you would have three h w so we have the volume b which is supposed to three h w that is b third equation and now we put equation two into equation three that means we have the volume which is equals to three in place of so let's have three w then in place of h we put h minus three w all on three plus w there so if we we multiply through we have 24 w minus 9 w squared all on three plus w all right okay.
04:35
So at this point we would actually this and you will take the first derivative of vary respect to w and equal to zero so that we can find the value of w which maximizes the area after that you will substitute the value of w into here and find the value of h so let's clear some portions and move on...