an open box with a volume of 1500cm cubed is to be constructed by taking a piece of cardboard of 20 cm by 40 cm, cutting squares of sides length x cm from each corner and folding up the sides. show that this can be done in two ways, and find exact dimensions of the box in each case.

To solve this problem, we need to find the dimensions of the box by cutting squares from each corner of the cardboard. We are given the cardboard dimensions of 20 cm by 40 cm and the volume of the box is 1500 cm³.

Let's consider the first case:

1. Cutting squares of side length x cm from each corner:
- After cutting the squares, the dimensions of the cardboard will be (40 - 2x) cm by (20 - 2x) cm.
- Folding up the sides will create a box with a length of (40 - 2x) cm, a width of (20 - 2x) cm, and a height of x cm.
- The volume of the box is given by length x width x height, which in this case is (40 - 2x) cm * (20 - 2x) cm * x cm.
- Setting the volume equal to 1500 cm³, we have the equation: (40 - 2x)(20 - 2x)x = 1500.
- We can solve this equation to find the value(s) of x.

Now, let's consider the second case:

2. Cutting squares of side length x cm from each corner:
- After cutting the squares, the dimensions of the cardboard will be (20 - 2x) cm by (40 - 2x) cm (note the interchange of dimensions).
- Folding up the sides will create a box with a length of (20 - 2x) cm, a width of (40 - 2x) cm, and a height of x cm.
- The volume of the box is given by length x width x height, which in this case is (20 - 2x) cm * (40 - 2x) cm * x cm.
- Again, setting the volume equal to 1500 cm³, we have the equation: (20 - 2x)(40 - 2x)x = 1500.
- This equation will also give us the value(s) of x.

By solving the equations for x in both cases, we can find the exact dimensions of the box.

To solve this problem, we need to follow the given steps:

Step 1: Let's assume that the length of the squares cut from each corner is x cm.

Step 2: After cutting out the squares from each corner, the dimensions of the resulting box will be (40 - 2x) cm by (20 - 2x) cm by x cm.

Step 3: The volume of the box is given by the formula:

Volume = Length x Width x Height

Substituting the known values, we have:

1500 = (40 - 2x) x (20 - 2x) x x

Step 4: Simplify the equation and express it in standard form:

1500 = (800 - 120x + 4x^2) x

Step 5: Expanding the equation further:

1500 = 4x^3 - 120x^2 + 800x

Step 6: Rearranging the equation and setting it equal to zero:

4x^3 - 120x^2 + 800x - 1500 = 0

Step 7: Now we need to solve this equation to find the values of x.

To find the exact dimensions of the box in each case, we need to solve the equation using factoring or other methods.

Cutting an x-cm notch from each corner of the sheet, leaves a box (20-2x)x(40-2x). We want that volume to be 1500.

(20-2x)(40-2x) = 1500
4x^2 - 120x - 700 = 0
x = -5 or 35

so, a 30x50 box will fit the bill

The solution x=35 produces a box of negative dimensions.