# Calculus

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.

1. 👍
2. 👎
3. 👁
1. 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)

1. 👍
2. 👎
2. 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!!!

1. 👍
2. 👎
3. 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!

1. 👍
2. 👎

## Similar Questions

1. ### 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

2. ### math

An open box is made from a rectangular piece of cardboard measuring 16 cm by 10cm. Four equal squares are to be cut from each corner and flaps folded up. Find the length of the side of the square which makes the volume of the box

3. ### 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.

4. ### college algebra

An open box is made from a square piece of cardboard 20 inches on a side by cutting identical squares from the corners and turning up the sides.(a) Express the volume of the box, V , as a function of the length of the side of the

1. ### calculus

By cutting away identical squares from each corner of a rectangular piece of cardboard and folding up the resulting flaps, an open box may be made. If the cardboard is 14 in. long and 6 in. wide, find the dimensions of the box

2. ### math

a 5cm by 5cm square is cut from each corner of a rectangular piece of cardboard.the sides are folded up to make an open box with a maximum volume.if the perimeter of the base is 50cm,what are the dimensions of the box.

3. ### Calculus 1

A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 18 in. by 30 in. by cutting out equal squares of side x at each corner and then folding up the sides as in the figure. Express the

4. ### calculus

An open box is to be made out of a 10-inch by 14-inch piece of cardboard by cutting out squares of equal size from the four corners and bending up the sides. Find the dimensions of the resulting box that has the largest volume.

1. ### math

a box is to be made by cutting out the corners of a square piece of cardboard and folding the edges up. if 3-inch squares are to be cut out of the corners and the box contains 243 cubic inches, what is the length of a side of the

2. ### Precalculus

A box with a square base and no top is to be made from a square piece of carboard by cutting 7 in. squares from each corner and folding up the sides. The box is to hold 1008 in3. How big a piece of cardboard is needed?

3. ### Basic Calculus

A piece of cardboard measuring 14 inches by 8 inches is formed into an open-top box by cutting squares with side length x from each corner and folding up the sides. FIND: 1). a formula for the volume of the box in terms of x 2).

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