- 27

- 0

**[SOLVED] Minimizing the Surface Area**

**1. Homework Statement**

A box has a bottom with one edge 8 times as long as the other. If the box has no top and the volume is fixed at

*V*, what dimensions minimize the surface area?

**2. Homework Equations**

*V*=

*lwh*

*SA*(with no top) =

*lw*+ 2

*lh*+ 2

*wh*

**3. The Attempt at a Solution**

*l*=

*x*

*w*= 8

*x*

*h*=

*V*/(8

*x*^2)

Finding an equation for the surface area.

*SA*=

*lw*+ 2

*lh*+ 2

*wh*

*SA*=

*x*(8

*x*) + 2

*x*(

*V*/(8

*x*^2)) + 2(8

*x*)(

*V*/(8

*x*^2))

*SA*= 8

*x*^2 +

*V*/(4

*x*) + 2

*V*/

*x*

Finding the derivative of the equation in order to set it equal to zero to find the critical points, so the minimum can be found.

(

*d*

*SA*)/(

*d*

*x*) = 16

*x*-

*V*/(4

*x*^2) - 2

*V*/(

*x*^2)

(

*d*

*SA*)/(

*d*

*x*) = (64

*x*^3 - 9

*V*) / (4

*x*^2)

(

*d*

*SA*)/(

*d*

*x*) = 0

(64

*x*^3 - 9

*V*) / (4

*x*^2) = 0

(64

*x*^3 - 9

*V*) = 0

64

*x*^3 = 9

*V*

*x*^3 = (9

*V*)/64

*x*= ((9

*V*)/64)^(1/3)

Plugging the solution into the equations for the dimensions.

*l*=

*x*= ((9

*V*)/64)^(1/3)

*w*= 8

*x*= 8((9

*V*)/64)^(1/3)

*h*=

*V*/(8

*(((9*^2)

*V*)/64)^(1/3))

**I am unsure if I did the right steps in order to find the solution.**

Also, I am not very confident in the work I did for each step.Also, I am not very confident in the work I did for each step.