AN OPEN BOX IS FORMED BY CUTTING SQUARES OUT OF A PIECE OF CARDBOARD THAT IS 16 FT BY 19 FT AND FOLDING UP THE FLAPS. WHAT SIZE CORNER SQUARES SHOULD BE CUT TO YEILD A BOX THAT HAS A VOLUME OF 175 CUBIC FEET

Let the side of the square to be cut out be x ft.

The volume of the resulting box is then
V = x(16-2x)(19-2x) = 175 ft³
You can solve the equation by trial and error, there should be a root around x=0.68 ft.

V = x(16-2x)(19-2x) = 175 ft³

Ok, thanks. Now I'm trying to multiply out and having some difficulty. I am getting 4x^3-70x^2+304x = 175 which I'm not sure how to work with

See response of follow-up post:

http://www.jiskha.com/display.cgi?id=1252257020

2.885

To find the size of the corner squares to cut in order to yield a box with a volume of 175 cubic feet, we need to follow these steps:

Step 1: Visualize the Box
First, let's visualize the open box formed from the cardboard. The top and bottom of the box will have the dimensions 16 ft by 19 ft. The height of the box will be h, and the corner squares will be cut out of each corner of the cardboard. When the flaps are folded up, those corner squares will form the sides of the box.

Step 2: Determine the Dimensions of the Box
The dimensions of the box without the corner squares will be (16 - 2x) ft by (19 - 2x) ft, where x is the length of the side of the corner square that will be cut out.

Step 3: Calculate the Volume of the Box
The volume of the box is calculated by multiplying its length, width, and height. In this case, the volume is given as 175 cubic feet. So, we have the equation:

V = (16 - 2x)(19 - 2x)h = 175

Step 4: Solve the Equation for x
We can rearrange the equation to solve for x in terms of h:

2x = 16 - (175 / h)^(1/2)

Step 5: Substitute the Value of x into the Equation
Now, substitute the previous expression for 2x back into the equation for the dimensions of the box:

(16 - 2x)(19 - 2x)h = 175

Step 6: Solve for h
Simplify and solve the equation for h:

(16 - 2x)(19 - 2x)h = 175

Expand the brackets:

(16 * 19 - 2x * 19 - 2x * 16 + 4x^2)h = 175

Combine like terms:

(304 - 38x - 32x + 4x^2)h = 175

Simplify:

(4x^2 - 70x + 304)h = 175

Divide both sides by h:

4x^2 - 70x + 304 = 175

Rearrange to form a quadratic equation:

4x^2 - 70x + 129 = 0

Step 7: Solve the Quadratic Equation
Solve the quadratic equation using factoring, completing the square, or the quadratic formula. In this case, the equation doesn't factor nicely, so we will use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

For the equation 4x^2 - 70x + 129 = 0, the coefficients are:
a = 4, b = -70, c = 129

Substitute these values into the quadratic formula and calculate the two possible values for x.

Step 8: Determine the Side Length of the Corner Square
Since x represents the side length of the corner square, you'll need to choose the positive value for x that makes sense in the context of the problem. Remember that the corner squares can't have negative side lengths.

By following these steps, you should be able to find the size of the corner squares to cut in order to achieve a box volume of 175 cubic feet.