Problem:
From a square of tin that measures 12inches on a side of a rectangular box is made by cutting out a small squares of the same size from the four corners and turning up the sides. If the volume of the box is 108 cubic inches ,what should be the length of the side of the square to be cut out?
v = x * (12-2x)(12-2x) =108
4 x^3 - 48 x^2 +144 x - 108 = 0
try x = 3
To solve this problem, we can use the given information and the formula for volume of a rectangular box (V = lwh) to find the length of the side of the square to be cut out.
Let's break down the problem into steps:
Step 1: Determine the width and height of the rectangular box.
Since the original tin is a square with side length 12 inches, the width and height of the rectangular box will also be 12 inches.
Step 2: Determine the length of the rectangular box.
Let's assume the length of the square to be cut out is "x" inches. Since the square is cut out from all four corners, the length of the rectangular box will be the original side length minus twice the length of the cut-out square. Therefore, the length of the rectangular box will be (12 - 2x) inches.
Step 3: Calculate the volume of the box.
The volume of the box is given as 108 cubic inches. We can calculate the volume using the formula V = lwh. In this case, V = 108 cubic inches, l = (12 - 2x) inches, w = 12 inches, and h = (12 - 2x) inches.
So, the equation for the volume becomes:
108 = (12 - 2x) * 12 * (12 - 2x)
Step 4: Solve the equation for x.
We can solve the equation for x by simplifying and rearranging it.
First, expand the equation:
108 = (144 - 24x - 24x + 4x^2)
108 = 144 - 48x + 4x^2
Rearrange the equation in standard form:
4x^2 - 48x + 36 = 0
Step 5: Solve the quadratic equation.
Now, we can solve the quadratic equation using factoring, completing the square, or the quadratic formula.
Since this equation can be easily factored, let's factor it:
4(x^2 - 12x + 9) = 0
4(x - 3)(x - 3) = 0
(x - 3)^2 = 0
From this, we can see that the only solution for x is 3.
Step 6: Determine the length of the side of the square to be cut out.
Since we found x = 3, the length of the side of the square to be cut out is 3 inches.
To solve this problem, we need to find the length of the side of the square that is cut out from each corner.
Let's assume that the side length of the square to be cut out is 'x' inches.
We know that the original square tin has a side length of 12 inches.
When we cut out a square of side length 'x' from each corner, the resulting length of the box will be 12 - 2x inches (since we cut out the square from both ends of each side).
Similarly, the resulting width of the box will also be 12 - 2x inches.
Finally, the height of the box is equal to the side length of the square that was cut out, which is 'x' inches.
To find the volume of the box, we multiply the length, width, and height:
Volume = Length x Width x Height
Now, we can solve for the value of 'x' using the given volume of 108 cubic inches:
108 = (12 - 2x) * (12 - 2x) * x
Simplifying the equation:
108 = (144 - 48x + 4x^2) * x
Rearranging terms:
108 = 4x^3 - 48x^2 + 144x
Dividing both sides by 4:
27 = x^3 - 12x^2 + 36x
Now we can see that 'x' must be a factor of 27. By trying different values, we can find that 'x' equals 3 satisfies the equation:
27 = 3^3 - 12(3^2) + 36(3)
27 = 27 - 108 + 108
Therefore, the length of the side of the square to be cut out should be 3 inches.