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 16128 in3. How big a piece of cardboard is needed?

?in by ?in

base area = (w-14)(w-14)

h = 7
volume = 7 (w-14)^2 = 16128

w^2-28w+196 = 2304

w^2 - 28 w - 2108 = 0

(w-62)(w+34) = 0

w = 62

To find out how big a piece of cardboard is needed to make the box, we need to determine the dimensions of the square base.

Let's denote the side length of the square base as "s", and the height of the box as "h".

Since 7-inch squares are cut from each corner of the cardboard, the dimensions of the base will be reduced by 14 inches (7 inches on each side).

Thus, the length and width of the base will be (s - 14) inches.

The height of the box will be the size of the square cut from each corner, which is 7 inches.

The volume of the box is given as 16128 in^3.

To find the dimensions of the base, we can set up an equation using the volume formula:

Volume = Base Area * Height

16128 = (s - 14)^2 * 7

Now, let's solve this equation step-by-step:

1. Divide both sides of the equation by 7:

2304 = (s - 14)^2

2. Take the square root of both sides to eliminate the square:

√2304 = √(s - 14)^2

48 = s - 14

3. Add 14 to both sides of the equation:

48 + 14 = s - 14 + 14

62 = s

Therefore, the side length of the square base is 62 inches.

To find the dimensions of the cardboard needed, we add 14 inches to each side:

Length = s + 14 = 62 + 14 = 76 inches

Width = s + 14 = 62 + 14 = 76 inches

So, a piece of cardboard measuring 76 inches by 76 inches is needed.

To determine the size of the cardboard needed, we can start by finding the dimensions of the square base of the box.

Let's assume the side length of the square base is "x" inches. Since 7-inch squares are cut from each corner, the resulting length of the base will be reduced by 14 inches (7 inches cut from each side). Therefore, the length of the base can be expressed as (x - 14) inches.

Similarly, the height of the box will be 7 inches since that is the amount folded up from each side.

To find the volume of the box, we multiply the dimensions of the base and height: Volume = (x - 14) * (x - 14) * 7

We know that the volume of the box is given as 16128 in³. So we can set up the equation:

(x - 14) * (x - 14) * 7 = 16128

Next, we can solve this equation to find the value of "x" and determine the dimensions of the square base.

Once we have the value of "x", we can find the size of the cardboard needed. The size of the cardboard would be the dimensions of the base plus the 7 inches folded up on each side.

Therefore, the size of the cardboard needed would be (x + 7) inches by (x + 7) inches.

side of base = x-14

height = 7

solve
7(x-14)^2 = 16128
(x-14)^2 = 2304

I will let you finish it.