Part 1 of 3 o points 0 of 5 let x be the length of one side...

Part 1 of 3 O Points: 0 of 5 Let x be the length of one side of the base of the box. Express the total cost of the materials, C, in terms of x. Save A worker is to construct an open rectangular box with a square base and a volume of 192 ft. If material for the bottom costs $6/f and material for the sides costs $8/f, what dimensions will result in the least expensive box? What is the minimum cost?

Transcript

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

00:04 Let me just draw that situation in here.

00:09 We have that box and we have an square base and it is open, so we don't need to consider the up and top of these box.

00:24 Now, the volume for this is 100 cubic feet.

00:28 Now the material for the bottom of this costs $8 per square foot, and the material for the sides costs $5 per square foot.

00:38 Now let's label the dimensions of this as, now since the base is a square, we're gonna label its dimensions as x and x, and the height of this we're gonna label just as h.

00:51 So then the volume for this is just x squared times the height that is h.

00:56 So we have that.

00:58 Now let's write the cost function first, because the question for this is what dimensions will result, so we need to determine the value of x and the height, and also that minimum cost.

01:09 So let's write the cost function.

01:10 The cost function is the area for the specific side times the price for that.

01:14 Let's start with the base of this.

01:16 The price is x and the area for that is x squared plus, we have four sides with an area of x times the height and the price for that is five.

01:24 So let's simplify things in here.

01:25 That will be a times x squared plus 20 times x times the height.

01:30 Now from the volume, we're gonna solve for the height.

01:33 So that will be 100 divided by x squared.

01:36 And then we substitute that into here.

01:38 So the cost function is a times x squared, and this plus we will have 20 times 100, and that will give us 2000 and that divided by x.

01:56 Right, now, once we have this, we differentiate this with respect to x in order to minimize these functions that will give us 16 times x.

02:03 Well, this is the function in terms of x, okay? if you need that.

02:07 So now we need to minimize the function with the area by this square with respect to x...

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