From a 18 inch by 18 inch piece of metal, squares are cut out of

Question

From a 18 inch by 18 inch piece of metal, squares are cut out of each of the four corners so that the sides can then be folded up to make a box. Let x represent the length of the sides of the squares, in inches, that are cut out.

 

A.) Express the volume as a function of the width

 

 

B). Give the domain of the volume function

 

C. Find the maximum volume.

Answer 1

From a 18 inch by 18 inch piece of metal, squares are cut out of each of the four corners so that the sides can then be folded up to make a box. Let x represent the length of the sides of the squares, in inches, that are cut out.

A.) Express the volume as a function of the width
V=(18-2•x) ² x 

 

I would call x  the height but I think this is what they want

 

B). Give the domain of the volume function

 

x can range from 0 to 9 inches

 

9 is 1/2 of 18

 

 

Note V(0)=0 and V(9)=0

 

C. Find the maximum volume.

 

V=(18-2•x) ² x

V=4x³-72x² +324x

 

V'=12x²-144x +324

 

The extrema are when V' =0

 

0=12x²-144x +324

0=x²-12x+27

0=(x-9)(x-3)

x=9in, 3 in

V(9)=0 in³ 

V(3)=12•12•3=432 in³ 

 

You have to look at V'' to determine if these are min or max

 

V''=24x-144

V''(9)=72 since positive 9 is local minimum

 

V''(3)=-72 since negative 3 is local Maximim

 

Calculate V at the domain end points 0, 9, and at these extrema 3,9

 

V(3)=432 in³

V(0)=0 in³

V(9)=0 in³

 

the maximum volume is when x=3

V(3)= 432 in³