Finding min. value of dimensions for a tank of water

[BOAS]

Hello

Homework Statement

An open tank is constructed with a square base and vertical sides to hold 32m³ of water. Find dimensions of the tank if the area of the sheet metal used to make it is to have a minimum value.

Homework Equations

The Attempt at a Solution

I'm not entirely sure of how to approach this problem beyond needing to use the second derivative. I think I need to construct an expression of the area related to volume.

So,

the sides pf the base, x multiply together to give an area x² and the four sides can be called 4xh (side of the base multiplied by height)

x² + 4xh is an area of sheet metal

x²h = 32

I don't know how to proceed.

Thank you for any help you can offer

 

[Mark44]

Hello

Homework Statement

An open tank is constructed with a square base and vertical sides to hold 32m³ of water. Find dimensions of the tank if the area of the sheet metal used to make it is to have a minimum value.

Homework Equations

The Attempt at a Solution

I'm not entirely sure of how to approach this problem beyond needing to use the second derivative. I think I need to construct an expression of the area related to volume.

So,

the sides pf the base, x multiply together to give an area x² and the four sides can be called 4xh (side of the base multiplied by height)

x² + 4xh is an area of sheet metal

x²h = 32

I don't know how to proceed.

Thank you for any help you can offer

The only thing you're missing is that your volume function can be solved for h as a function of x, and then substituted into your surface area function, making it a function of x alone. Once you have the surface area as a function of x, use the usual technique for finding the minimum value.

 

[BOAS]

The only thing you're missing is that your volume function can be solved for h as a function of x, and then substituted into your surface area function, making it a function of x alone. Once you have the surface area as a function of x, use the usual technique for finding the minimum value.

ah of course! Thanks

 

[Ray Vickson]

Hello

Homework Statement

An open tank is constructed with a square base and vertical sides to hold 32m³ of water. Find dimensions of the tank if the area of the sheet metal used to make it is to have a minimum value.

Homework Equations

The Attempt at a Solution

I'm not entirely sure of how to approach this problem beyond needing to use the second derivative. I think I need to construct an expression of the area related to volume.

So,

the sides pf the base, x multiply together to give an area x² and the four sides can be called 4xh (side of the base multiplied by height)

x² + 4xh is an area of sheet metal

x²h = 32

I don't know how to proceed.

Thank you for any help you can offer

Besides the hint already given, you can also use the Lagrange multiplier method (if you know about it).

 

[BOAS]

Besides the hint already given, you can also use the Lagrange multiplier method (if you know about it).

I don't know about that.

I'm in a physics foundation year (year 0 at my uni) due to not having done the traditional subjects that lead to taking a physics degree.

 

[haruspex]

There is a shortcut available here. If the tank also had a top, you'd straight away say it should be a cube. With the open tank, imagine creating a closed tank by taking two optimised open tanks and inverting one over the other. You must now have created an optimal closed tank, i.e. a cube.