Calculus

[abihoward]

Use the method of Lagrange Multipliers to solve: An aquarium is to be built in the shape of a rectangular prism with an open top. the material to make the bottoom surface costs 3 times as much per square foot as the material to make the 4 sides surfaces. The volume of the aquarium is to be 5 cubic feet . Find the dimension of the aquarium that minimizes the cost of the surfaces. Give your
answer both an exact form and rounded to two decimal places.

Can any one help me in this regards.
Sincerely yours

abi

 

[Deleted member 4993]

Use the method of Lagrange Multipliers to solve: An aquarium is to be built in the shape of a rectangular prism with an open top. the material to make the bottoom surface costs 3 times as much per square foot as the material to make the 4 sides surfaces. The volume of the aquarium is to be 5 cubic feet . Find the dimension of the aquarium that minimizes the cost of the surfaces. Give your
answer both an exact form and rounded to two decimal places.

Can any one help me in this regards.
Sincerely yours

abi

First define your variables - what do you need to find - assign variable names (such as L for length) to those.

Then write your cost function and constraints according to those variables.

Please show us your work, indicating exactly where you are stuck - so that we may know where to begin to help you.

 

[galactus]

Here is a Lagrange multiplier. This should help you tackle other problems like this.

The surface area is what must be minimized.

The surface area is given by \(\displaystyle S=xy+2xz+2yz\)

The cost is given by \(\displaystyle 3Cxy+C(2xz+2yz)\)

Subject to the constraint \(\displaystyle xyz=5\)

If we let \(\displaystyle f(x,y,z)=3Cxy+C(2xz+2yz)\) and \(\displaystyle g(x,y,z)=xyz\), then

\(\displaystyle {\nabla}f(x,y,z)=C(3y+2z)i+C(3x+2z)j+2C(x+y)k\)

and

\(\displaystyle {\nabla}g=yzi+xzj+xyk\)

\(\displaystyle {\nabla}g\neq 0\) at any point on the surface xyz=5, since x, y, and z are all non-zero.

So, we have \(\displaystyle {\nabla}f={\lambda}{\nabla}g\)

\(\displaystyle C(3y+2z)={\lambda}yz\)..........[1]

\(\displaystyle C(3x+2z)={\lambda}xz\)...........[2]

\(\displaystyle 2C(x+y)={\lambda}xy\)..........[3]

From [1], \(\displaystyle {\lambda}=\frac{2C}{y}+\frac{3C}{z}\)

From [2], \(\displaystyle {\lambda}=\frac{2C}{x}+\frac{3C}{z}\)

From [3], \(\displaystyle {\lambda}=\frac{2C}{x}+\frac{2C}{y}\)

From the first two equations:

\(\displaystyle x=y\)

From the second and third:

\(\displaystyle z=\frac{3x}{2}\)

Sub these into the volume constraint and we get:

\(\displaystyle x^{2}(\frac{3x}{2})=5\)

The dimensions are easily obtained from here.