We are given a rectangular piece of sheet metal with the width being 50 cm shorter than the length. Squares of 10 cm by 10 cm are cut from each corner, and the remaining flaps are folded up to form an open box. The volume of this box is 104,000 cm³, and we need to determine the length and width of the original sheet metal.

Step 1: Define Variables

Let:

• $L$ be the length of the rectangular sheet.
• $W$ be the width of the rectangular sheet.

From the problem, we know that:

• $W = L - 50$ (since the width is 50 cm shorter than the length).

Step 2: Dimensions of the Box

After cutting out the 10 cm squares from each corner and folding the sides up:

• The new length of the base of the box will be $L - 20$ (since 10 cm is cut from both sides).
• The new width of the base of the box will be $W - 20$.
• The height of the box will be 10 cm (from the squares cut out and folded up).

Step 3: Volume of the Box

The volume of the box is given by the formula: $V = \text{Length of base} \times \text{Width of base} \times \text{Height}$ Substituting the known values: $104,000 = (L - 20)(W - 20) \times 10$ $104,000 = 10(L - 20)(W - 20)$ Dividing both sides by 10: $10,400 = (L - 20)(W - 20)$

Step 4: Substitute $W = L - 50$

Now substitute $W = L - 50$ into the equation: $10,400 = (L - 20)((L - 50) - 20)$ Simplify the expression inside the parentheses: $10,400 = (L - 20)(L - 70)$

Step 5: Expand and Solve the Quadratic Equation

Now expand the right-hand side: $10,400 = L^2 - 70L - 20L + 1,400$ $10,400 = L^2 - 90L + 1,400$ Move everything to one side to form a quadratic equation: $L^2 - 90L + 1,400 - 10,400 = 0$ $L^2 - 90L - 9,000 = 0$

Step 6: Solve the Quadratic Equation

We will now solve the quadratic equation: $L^2 - 90L - 9,000 = 0$ Using the quadratic formula: $L = \frac{-(-90) \pm \sqrt{(-90)^2 - 4(1)(-9,000)}}{2(1)}$ $L = \frac{90 \pm \sqrt{8,100 + 36,000}}{2}$ $L = \frac{90 \pm \sqrt{44,100}}{2}$ $L = \frac{90 \pm 210}{2}$

This gives us two possible solutions for $L$: $L = \frac{90 + 210}{2} = 150 \quad \text{or} \quad L = \frac{90 - 210}{2} = -60$ Since the length cannot be negative, we have $L = 150$ cm.

Step 7: Calculate the Width

Now that we know $L = 150$ cm, we can find the width: $W = L - 50 = 150 - 50 = 100 \text{ cm}$

Final Answer:

• The length of the original piece of sheet metal is 150 cm.
• The width of the original piece of sheet metal is 100 cm.

Would you like a detailed explanation on any part of this solution, or do you have any other questions?

Related Questions:

1. What is the formula for the volume of a rectangular prism?
2. How do you solve a quadratic equation using the quadratic formula?
3. How does cutting out squares affect the dimensions of the base of the box?
4. What other shapes could be cut to form different open boxes?
5. How does increasing the size of the cut squares affect the volume of the resulting box?

Tip: Always double-check units when working through problems involving geometry and volume calculations to avoid mix-ups!