To solve this problem, we need to follow these steps:
Step 1: Find the dimensions of the square to be cut from the corners.
Let's assume the side length of the square to be cut from each corner is "x" cm.
Step 2: Calculate the dimensions of the box base.
After cutting the square and folding up the resulting flaps, the base of the box will have dimensions:
Length = 12 cm - 2x
Width = 8 cm - 2x
Height = x
Step 3: Calculate the volume of the box.
The volume of the box is given as 36 cm³:
Volume = Length × Width × Height
36 = (12 - 2x) × (8 - 2x) × x
Step 4: Solve the equation.
Let's solve the equation to find the value of "x".
36 = (12 - 2x) × (8 - 2x) × x
36 = (96 - 36x - 16x + 4x²) × x
36 = (4x² - 52x + 96) × x
36 = 4x³ - 52x² + 96x
Step 5: Simplify the equation.
Bring the equation to standard form:
4x³ - 52x² + 96x - 36 = 0
Step 6: Factor the equation.
After factoring, we get:
(x - 3)(x - 4)(4x - 3) = 0
Step 7: Find the values of "x" that satisfy the equation.
From the factored equation, we have three possible solutions:
x = 3, x = 4, and x = 3/4 (0.75).
Step 8: Calculate the dimensions of the box.
Using the values of "x" we found, we can calculate the dimensions of the box as follows:
For x = 3:
Length = 12 - 2(3) = 6 cm
Width = 8 - 2(3) = 2 cm
Height = 3 cm
For x = 4:
Length = 12 - 2(4) = 4 cm
Width = 8 - 2(4) = 0 cm (not possible)
Height = 4 cm
For x = 3/4:
Length = 12 - 2(3/4) = 10.5 cm (not an integer)
Therefore, the integer dimensions of the box are:
Length = 6 cm
Width = 2 cm
Height = 3 cm.