To solve this problem, we can follow these steps:
Step 1: Visualize the problem
Imagine a rectangular sheet of material where squares measuring 6cm by 6cm are removed from each corner. When the remaining material is folded, it forms an open-top box with a height of 6cm.
Step 2: Understand the given information
From the problem, we know that the rectangular sheet has an area of 480 cm², and the squares cut from each corner measure 6cm by 6cm. The height of the box is given as 6cm.
Step 3: Define the variables
Let's assign variables to the length and width of the box to solve for them. Let's say:
- Length of the box = L cm
- Width of the box = W cm
Step 4: Set up an equation for the area
The area of the rectangular sheet can be computed by multiplying the length (L + 2 * 6) and the width (W + 2 * 6):
(L + 2 * 6) * (W + 2 * 6) = 480
Step 5: Simplify the equation
Simplify the equation by distributing and combining like terms:
(L + 12) * (W + 12) = 480
LW + 12L + 12W + 144 = 480
LW + 12L + 12W = 480 - 144
LW + 12L + 12W = 336
Step 6: Substitute the relationship between length and width
We know from the given information that the length of the box is 4 cm longer than its width. So, we can write the relationship as:
L = W + 4
Substitute this relationship into the equation we have:
(W + 4) * W + 12(W + 4) + 12W = 336
Step 7: Solve the equation
Expand and simplify the equation further:
W² + 4W + 12W + 48 + 12W = 336
W² + 28W + 48 = 336
W² + 28W - 288 = 0
Step 8: Solve the quadratic equation
To solve the quadratic equation, we can factor it or use the quadratic formula. In this case, let's use the quadratic formula:
W = (-b ± √(b² - 4ac))/(2a)
Where a = 1, b = 28, and c = -288:
W = (-28 ± √(28² - 4 * 1 * -288))/(2 * 1)
Simplify the equation:
W = (-28 ± √(784 + 1152))/2
W = (-28 ± √1936)/2
W = (-28 ± 44)/2
Now, we have two possible values for W:
1. W = (-28 + 44)/2 = 16/2 = 8
2. W = (-28 - 44)/2 = -72/2 = -36
Since width cannot be negative, we discard the second value.
Step 9: Calculate the length
Since we know L = W + 4, we can substitute W = 8 into this equation:
L = 8 + 4 = 12
Step 10: Determine the dimensions of the box
The dimensions of the box are:
- Length = L = 12 cm
- Width = W = 8 cm
Therefore, the length and width of the box are 12 cm and 8 cm, respectively.