# Calculus

posted by .

A box with an open top is to be made from a square piece of cardboard by cutting equal squares from the corners and turning up the sides. If the piece of cardboard measures 12 cm on the side, find the size of the squares that must be cut out to yield the maximum volume of the box.

• Calculus -

Let the length of the (four) squares cut from the corner be x.
The box will then have dimensions 12-2x, 12-2x and x once the open box is made.
The volume is therefore:
V(x)=x(12-2x)²
Differentiate with respect to x and equate to zero to get the greatest voume:
V'(x)=(12-2x)²+2x(12-2x)(-2)
=12(x²-8)
Equate to zero and solve for x:
12(x²-8)=0
=>
x=sqrt(8)

• Calculus -

Okay, so im not understanding how you went from V'(x)=(12-2x)²+2x(12-2x)(-2) to V'(x)=12(x²-8)... ive done this problem many times and am stuck on this part. i always end up with V'(x)= 12x²-96x+144 => 12(x²-8x+12)
so i end up with x=2 and x=6... which doesn't make sense because x=6 would make the base 0??? Help please!!!

• Calculus -

Trisha, your answer is correct; x=2 or 6. However this is where we have to use intuition and realize that x=6 is indeed extraneous (since a base of 0 will not be a box). Therefore x=2 is the answer!

## Similar Questions

1. ### Algebra

A box with a square base and no top is to be made from a square piece of cardboard by cutting 4-in squares from each corner and folding up the sides... the box is told 100 in cube, how big a piece of cardboard is needed?
2. ### calculus

An open box is to be made from a square piece of cardboard, 32 inches on a side, by cutting equal squares with sides of length x from the corners and turning up the sides (see figure below). Determine the function, V, in terms of x, …
3. ### calculus

An open box of maximum volume is to be made from a square piece of cardboard, 24 inches on each side, by cutting equal squares from the corners and turning up the sides to make the box. (a) Express the volume V of the box as a function …
4. ### Calculus

an open box is made by cutting out squares from the corners of a rectangular piece of cardboard and then turning up the sides. If the piece of cardboard is 12 cm by 24 cm, what are the dimensions of the box that has the largest volume …
5. ### math

A box with no top is to be constructed from a piece of cardboard whose width measures x cm and whose length measures 6 cm more than its width. The box is to be formed by cutting squares that measure 2 cm on each side from the four …
6. ### calculus

An Open Box Of Maximum Volume Is To Be Made From A Square Piece Of Cardboard Twentyfour Inches On Each Side By Cutting Equal Suares Frm The Corners And Turning Up The Sides
7. ### Algebra

A box with no top is to be constructed from a piece of cardboard whose length measures 6 inch more than its width. The box is to be formed by cutting squares that measure 2 inches on each side from the four corners an then folding …
8. ### Engineering

A box with an open top is to be made from a square piece of cardboard by cutting equal squares from the corners and turning up the sides. If the piece of cardboard measures 12 in on the side, find the size of the squares that must …
9. ### Calculus

An open box is formed from a piece of cardboard 12 inches square by cutting equal squares out of the corners and turning up the sides, find the dimensions of the largest box that can be made in this way.
10. ### Math

An open box is formed from a piece of cardboard 12 inches square by cutting equal squares out of the corners and turning up the sides. Find the volume of the largest box that can be made. Help!

More Similar Questions