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Part 1

A machine produces open boxes using square sheets of metal.  The machine cuts​ equal-sized squares measuring

3

inches on a side from the corners and then shapes the metal into an open box by turning up the sides.  If each box must have a volume of

243

cubic​ inches, find the length and width of the open box.

To solve this problem, we need to find the length and width of the original square sheet of metal that will result in an open box with a volume of 243 cubic inches, after cutting out 3-inch squares from each corner.

### Step-by-step solution:

1. **Understanding the problem**:
   - A square sheet of metal has squares cut from each corner.
   - Each square has a side length of 3 inches.
   - After cutting, the metal is folded up to form an open box.
   - The volume of the box is given as 243 cubic inches.
   
2. **Variables**:
   Let \( x \) be the side length of the original square sheet of metal.
   After cutting 3-inch squares from each corner, the new dimensions of the base of the box will be:
   - **Length**: \( x - 2(3) = x - 6 \)
   - **Width**: \( x - 6 \)
   - **Height**: 3 inches (since this is the height formed by folding up the sides)

3. **Volume formula**:
   The volume \( V \) of a box is given by:
   \[
   V = \text{Length} \times \text{Width} \times \text{Height}
   \]
   Substituting the expressions for length, width, and height:
   \[
   243 = (x - 6)(x - 6)(3)
   \]
   
4. **Solve for \( x \)**:
   First, divide both sides of the equation by 3:
   \[
   81 = (x - 6)(x - 6)
   \]
   Simplify:
   \[
   81 = (x - 6)^2
   \]
   Take the square root of both sides:
   \[
   \sqrt{81} = x - 6
   \]
   \[
   9 = x - 6
   \]
   Solving for \( x \):
   \[
   x = 9 + 6 = 15
   \]

Thus, the side length of the original square sheet of metal is **15 inches**.

### Final Answer:
The length and width of the base of the open box are:
- Length: \( 15 - 6 = 9 \) inches
- Width: \( 15 - 6 = 9 \) inches

### The dimensions of the open box:
- Length = 9 inches
- Width = 9 inches
- Height = 3 inches

Would you like further details or have any questions?

### Related Questions:
1. How would the volume change if the height of the box is reduced to 2 inches?
2. What is the area of the original square sheet of metal?
3. If the volume were doubled, what would the new dimensions of the box be?
4. How would the dimensions change if the squares cut from the corners were 4 inches instead of 3 inches?
5. What would happen if the original metal sheet was a rectangle instead of a square?

### Tip:
Whenever solving for dimensions involving volumes, remember to express all sides in terms of the variable before substituting into the volume formula!

A machine produces open boxes using square sheets of metal. The machine cuts equal-sized squares measuring 3 inches on a side from the corners and then shapes the metal into an open box by turning up the sides. If each box must have a volume of 243 cubic inches, find the length and width of the open box.

Learn how to solve a real-world geometry problem where you find the dimensions of an open box made from a square metal sheet. This involves algebraic manipulation, factoring, and volume formulas. Suitable for students in Grades 9-11.

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