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, as shown in the figure. The box is to hold 196 in3. How big a piece of cardboard is needed?

15 by 15inches

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

Let's start by assuming that the side length of the square base is x inches.

When we cut 4-inch squares from each corner, the dimensions of the base reduce by 8 inches (4 inches from each side).
Therefore, the dimensions of the base become (x - 8) inches by (x - 8) inches.

To construct the box, we fold up the sides of the cardboard. Since there is no top to the box, the height will be equal to the 4-inch squares that were cut out.

The height of the box will be 4 inches.

The volume of the box is given as 196 cubic inches. We can calculate the volume using the formula:

Volume = Length × Width × Height

Substituting the known values:

196 = (x - 8) × (x - 8) × 4

Let's solve this equation step by step:

1. Divide both sides of the equation by 4:
49 = (x - 8) × (x - 8)

2. Take the square root of both sides of the equation:
√49 = √((x - 8) × (x - 8))
7 = x - 8

3. Add 8 to both sides of the equation:
7 + 8 = x - 8 + 8
15 = x

The side length of the square base, x, is equal to 15 inches.

Now, to determine the size of the cardboard needed, we need to consider that the base dimensions will be (x - 8) inches by (x - 8) inches.

Substituting the value of x:
Base dimensions = (15 - 8) inches by (15 - 8) inches
Base dimensions = 7 inches by 7 inches

To determine the size of the cardboard, we add the dimensions of the base to the height of the box:
Cardboard size = Base length + height by Base width + height
Cardboard size = 7 inches + 4 inches by 7 inches + 4 inches
Cardboard size = 11 inches by 11 inches

Therefore, a square piece of cardboard measuring 11 inches by 11 inches is needed to create the box.

To determine the size of the cardboard needed, we can work backwards from the given volume of the box.

1. Start by visualizing the box formed by folding up the sides after cutting squares from the corners. The resulting shape resembles a rectangular prism or a "box" with an open top and a square base.

2. Let's assume that the side length of the square base is 'x'. Since we are cutting 4-inch squares from each corner, the resulting length along each side of the base will be reduced by 8 inches. Therefore, the base of the box will have dimensions (x - 8) inches by (x - 8) inches.

3. The height of the box will be the length of the 4-inch squares that were cut out from each corner. These squares have a height of 4 inches. Thus, the height of the box will be 4 inches.

4. Now that we know the dimensions of the box, we can calculate the volume. The formula for the volume of a rectangular prism is V = length * width * height. In this case, the volume is given as 196 in3, so we have the equation 196 = (x - 8)(x - 8)(4).

5. Simplify the equation by multiplying (x - 8)(x - 8) and multiplying the result by 4: 196 = 4(x - 8)(x - 8).

6. Divide both sides of the equation by 4 to isolate the multiplication term: 49 = (x - 8)(x - 8).

7. Take the square root of both sides to solve for x - 8: √49 = x - 8.

8. Simplify the square root of 49 to 7: 7 = x - 8.

9. Add 8 to both sides of the equation to solve for x: 7 + 8 = x.

10. x = 15.

11. Now we know that the side length of the square base is 15 inches. To find the size of the cardboard needed, we need to consider the dimensions of the unfolded cardboard. It will have the same length as the perimeter of the square base, so the size of the cardboard required is 4 * (15 + 4) = 4 * 19 = 76 inches.

Therefore, a piece of cardboard with dimensions of 76 inches by 76 inches is needed to make the desired box.