The problem involves finding the side length of the squares cut from each corner of a 10 cm by 90 cm piece of tin to form an open box, such that the area of the base of the resulting box is 516 cm².

Step-by-Step Solution:

1. Dimensions of the box's base: After cutting a square with side length $x$ from each corner, the dimensions of the base are:

   • Length = $90 - 2x$
   • Width = $10 - 2x$

2. Area of the base: The area of the base is given as $516 \, \text{cm}^2$. Using the formula for the area of a rectangle, we have: $(90 - 2x)(10 - 2x) = 516$

3. Expand and simplify: Expand the equation: $900 - 180x - 20x + 4x^2 = 516$ Combine like terms: $4x^2 - 200x + 900 = 516$ Simplify further: $4x^2 - 200x + 384 = 0$

4. Solve the quadratic equation: Divide through by 4 to simplify: $x^2 - 50x + 96 = 0$ Solve using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Here, $a = 1$, $b = -50$, and $c = 96$. Substituting: $x = \frac{-(-50) \pm \sqrt{(-50)^2 - 4(1)(96)}}{2(1)}$ $x = \frac{50 \pm \sqrt{2500 - 384}}{2}$ $x = \frac{50 \pm \sqrt{2116}}{2}$ $x = \frac{50 \pm 46}{2}$

5. Find the roots: $x = \frac{50 + 46}{2} = 48 \quad \text{or} \quad x = \frac{50 - 46}{2} = 2$

6. Choose the valid solution: Since $x = 48$ would make the dimensions of the tin negative, the valid solution is: $x = 2 \, \text{cm}$

Final Answer:

The side length of the squares is 2 cm.

Do you have any questions, or would you like me to explain any step in more detail?

Related Questions:

1. What would the dimensions of the box be after the cuts are made?
2. How would the formula change if the tin dimensions were different?
3. Can we solve this problem graphically?
4. How does changing the area of the base affect the dimensions of the squares?
5. How would you calculate the volume of the box formed?

Tip:

When solving quadratic equations, always verify the solution to ensure it satisfies the physical constraints of the problem.