V lxwxh 1000m3 sa 2lw 2lh 2wh to minimize the surface area...

V= lxwxh = 1000m³ SA = 2lw + 2lh + 2wh to minimize the surface area, the dimensions should be as close to a cube as possible because a cube has the smallest surface area for a given volume.

Transcript

00:01 Hi there, so for this problem we have a rectangular box.

00:04 So let me just draw this situation for this.

00:07 We have something like this, right? you have some depth for this box because it's in three dimensions.

00:17 Okay, something like that, okay? now, it has an up and top, so we don't need to consider the top for the surface area.

00:26 And it has a volume b, right? so this is the volume.

00:31 Now, we're gonna label the dimensions of the base as x and x because we are told that it has a square base.

00:40 And we're gonna label the height of this as h.

00:45 So then we know that the volume is going to be defined as the product between all of these dimensions.

00:50 So that will be then x times x times the height.

00:53 So that will be x squared times the height.

00:56 Now, the question is, find the dimensions, so that will be the value of x and the value of the height such that we will find, we'll minimize the surface area.

01:07 Now, the surface area is just the sum of all of the areas, but for these box.

01:13 So we start for the area for the base.

01:16 So that will be then x squared plus the remaining four sides of this, the area is just x times h.

01:26 So we have four sides with an area of x times h, just like this.

01:31 Now, we need to have, before we minimize this function, we need to have this function in terms of only one variable.

01:40 So from the volume, we are going to solve for x...

Question

V= lxwxh = 1000m³ SA = 2lw + 2lh + 2wh to minimize the surface area, the dimensions should be as close to a cube as possible because a cube has the smallest surface area for a given volume.

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