The volume of a box is the amount of space in the box
The dimensions that minimize the cost of the box is 4 in by 4 in by 4 in
<h3>How to determine the dimensions that minimize the cost</h3>
The dimensions of the box are:
Width = x
Depth = y
So, the volume (V) is:
![V = x^2y](https://tex.z-dn.net/?f=V%20%3D%20x%5E2y)
The volume is given as 64 cubic inches.
So, we have:
![x^2y = 64](https://tex.z-dn.net/?f=x%5E2y%20%3D%2064)
Make y the subject
![y = \frac{64}{x^2}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B64%7D%7Bx%5E2%7D)
The surface area of the box is calculated as:
![A =x^2 + 4xy](https://tex.z-dn.net/?f=A%20%3Dx%5E2%20%2B%204xy)
The cost is:
--- the base is twice as expensive as the sides
Substitute ![y = \frac{64}{x^2}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B64%7D%7Bx%5E2%7D)
![C = 2x^2 + 4x * \frac{64}{x^2}](https://tex.z-dn.net/?f=C%20%3D%202x%5E2%20%2B%204x%20%2A%20%5Cfrac%7B64%7D%7Bx%5E2%7D)
![C =2x^2 + \frac{256}{x}](https://tex.z-dn.net/?f=C%20%3D2x%5E2%20%2B%20%20%5Cfrac%7B256%7D%7Bx%7D)
Differentiate
![C' =4x - \frac{256}{x^2}](https://tex.z-dn.net/?f=C%27%20%3D4x%20-%20%20%5Cfrac%7B256%7D%7Bx%5E2%7D)
Set to 0
![4x - \frac{256}{x^2} = 0](https://tex.z-dn.net/?f=4x%20-%20%20%5Cfrac%7B256%7D%7Bx%5E2%7D%20%3D%200)
Multiply through by x^2
![4x^3 - 256= 0](https://tex.z-dn.net/?f=4x%5E3%20-%20%20256%3D%200)
Divide through by 4
![x^3 - 64= 0](https://tex.z-dn.net/?f=x%5E3%20-%20%2064%3D%200)
Add 64 to both sides
![x^3 = 64](https://tex.z-dn.net/?f=x%5E3%20%3D%20%2064)
Take the cube roots of both sides
![x = 4](https://tex.z-dn.net/?f=x%20%3D%204)
Recall that:
![y = \frac{64}{x^2}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B64%7D%7Bx%5E2%7D)
So, we have:
![y = \frac{64}{4^2}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B64%7D%7B4%5E2%7D)
![y = 4](https://tex.z-dn.net/?f=y%20%3D%204)
Hence, the dimensions that minimize the cost of the box is 4 in by 4 in by 4 in
Read more about volume at:
brainly.com/question/1972490