# You have to make a square-bottomed, unlidded box with a height of three inches and a volume of approximately 42 cubic inches. You will be taking a piece of cardboard, cutting three-inch squares from each corner, scoring between the corners, and folding up the edges. What should be the dimensions of the cardboard to the smartest quarter inch?

## Well, it seems like you're on a mission to create a box that's both smart and funny! Let's dive right into it. To determine the dimensions of the cardboard, we need to work backward a bit.

First, let's calculate the dimensions of the base of our box. Since we have to remove three-inch squares from each corner, it means that the original base of the box will be three inches smaller in both length and width than our final base. So, let's call the original length and width dimensions "L-6" and "W-6".

Now, let's consider the height of the box, which is given as three inches. Since that dimension is fixed, we don't have to modify it.

The volume of the box is given as approximately 42 cubic inches. To calculate the volume, we can use the formula: Volume = Length × Width × Height.

Substituting the values, we get:
42 = (L-6) × (W-6) × 3

Now, let's solve this equation to find the dimensions of the cardboard. However, since we are working with smart quarter inches, I'll let you take over the math part so you can flex those brain muscles! Good luck!

## To make a square-bottomed, unlidded box with a height of three inches and a volume of approximately 42 cubic inches, you will need to follow these steps:

Step 1: Determine the dimensions of the square base:
Since the volume of the box is calculated by multiplying the length, width, and height, and the height is given as 3 inches, we can calculate the area of the base by dividing the volume by the height:
Area of the base = Volume / Height
Area of the base = 42 cubic inches / 3 inches
Area of the base = 14 square inches

Since the base is a square, all sides are equal. Therefore, the length and width of the base are the same.

Step 2: Find the dimensions of the base.
Let's assume the side length of the square base is x inches.
The base area is calculated as x * x = x^2 square inches.
So, we have the equation x^2 = 14 square inches.

Step 3: Solve for x.
To find x, take the square root of both sides of the equation:
√(x^2) = √14
x = √14 inches
x ≈ 3.74 inches

Now that we have the side length of the square base, we can determine the dimensions of the original piece of cardboard.

Step 4: Calculate the dimensions of the cardboard.
Since we cut out three-inch squares from each corner, the length and width of the original piece of cardboard will be x + 6 inches.
Therefore, the dimensions of the cardboard would be approximately 9.74 inches by 9.74 inches (x + 6 ≈ 3.74 + 6 = 9.74 inches).

It is important to note that the dimensions are rounded to the smartest quarter inch, so for practical purposes, you may choose to round the dimensions to the nearest accurate quarter inch increment (e.g., 9.75 inches by 9.75 inches).

## To make a square-bottomed, unlidded box with a height of three inches and a volume of approximately 42 cubic inches, you will need to calculate the dimensions of the cardboard needed. Let's go step by step:

1. Start by visualizing the box. Since the box is square-bottomed, the length and width of the base will be the same, let's call it "x inches."

2. Given that you will be cutting three-inch squares from each corner of the cardboard, subtracting twice the height from both dimensions will give us the dimensions of the box's base. Thus, the dimensions of the base are (x - 2 * 3) inches, which simplifies to (x - 6) inches.

3. The height of the box is already given as 3 inches.

4. To find the volume of the box, multiply the dimensions of the base (length and width) by the height. In this case, we want the volume to be approximately 42 cubic inches, so we can write the equation as (x - 6)(x - 6)(3) ≈ 42.

5. Simplify the equation by expanding and collecting like terms: 3(x^2 - 12x + 36) ≈ 42.

6. Divide both sides of the equation by 3 to isolate the quadratic term: x^2 - 12x + 36 ≈ 14.

7. Rearrange the equation to make it equal to zero by subtracting 14 from both sides: x^2 - 12x + 22 ≈ 0.

8. To find the dimensions of the cardboard to the nearest quarter inch, we can solve the quadratic equation using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).

- In this case, a = 1, b = -12, and c = 22.
- Using these values, the quadratic formula gives two possible values for x: x ≈ 9.8151 or x ≈ 2.1849.

9. Since the dimensions of the cardboard cannot be negative, we consider only the positive value: x ≈ 9.8151 inches.

10. We need to round this number to the nearest quarter inch according to the problem. Rounding 9.8151 to the nearest quarter inch gives 9.75 inches.

Therefore, the dimensions of the cardboard should be approximately 9.75 inches by 9.75 inches to create a square-bottomed, unlidded box with a height of three inches and a volume of approximately 42 cubic inches.

## since the bottom is square, you will be using a square piece of cardboard.

So, if the uncut piece is of side x. after the cut, you will have volume

v = (x-6)^2 * 3

Now, you want v=42, so solve

3(x-6)^2 = 42
(x-6)^2 = 14
x = 6+√14

so, after cutting and folding, the volume will be √14*√14*3 = 42