a box is to be made by cutting out the corners of a square piece of cardboard and folding the edges up. if 3-inch squares are to be cut out of the corners and the box contains 243 cubic inches, what is the length of a side of the original cardboard square
let the base be x by x
after cut out, base is (x-3) by (x-3) and the height is 3
3(x-6)(x-6) = 243
(x-6)^2 = 81
x-6 = ±9
x = 15 or x is a negative which would make no sense
so the base was 15 by 15 inches
check:
after cutting out 3 inches from each end, the base is 9 by 9
volume = 9x9x3 = 243
Well, it looks like we have a bit of a puzzler here. So, let's get started.
Since we're cutting out 3-inch squares from each corner, we'll reduce the length and width of the cardboard square by 6 inches (2 corners * 3 inches). This means that the length of each side of the box will be the length of the original cardboard square minus 6 inches.
Now, let's move on to finding the volume of the box. We know that the volume of a rectangular box is given by the formula length × width × height. In this case, the length and width of the base of the box are the same, so we'll just use one variable, let's say "x," to represent the length of each side of the original cardboard square.
Now, we need to find the height of the box. Since we're folding up the edges of the cardboard, the height will be 3 inches.
So, the volume of the box is:
V = (x - 6) × (x - 6) × 3
Now we can plug in the given volume of 243 cubic inches and solve for x:
243 = (x - 6) × (x - 6) × 3
Simplifying this equation might take a bit, but don't worry, I'm here to entertain you while we do the math!
So, let's get our calculators out and have some fun! I hope you're ready for some math magic!
To find the length of a side of the original cardboard square, we can follow these steps:
1. Let's assume that the side length of the original square cardboard is "x" inches.
2. Since 3-inch squares are cut out from each corner, the length and width of the resulting box will be decreased by 6 inches (3 inches from each side). Therefore, the dimensions of the box will be (x - 6) inches.
3. As the cardboard is folded to create the box, the height of the box will be equal to the side length of the cutout squares, which is 3 inches.
4. To calculate the volume of the resulting box, we use the formula: Volume = length × width × height. Thus, the equation becomes:
(x - 6) × (x - 6) × 3 = 243
5. Now we can solve this equation for x. Let's simplify it:
3(x - 6)(x - 6) = 243
(x - 6)(x - 6) = 81
6. Taking the square root of both sides to eliminate the squared terms:
x - 6 = √81
x - 6 = 9
7. Solving for x:
x = 9 + 6
x = 15
Therefore, the length of a side of the original cardboard square is 15 inches.
To find the length of a side of the original cardboard square, we need to follow these steps:
Step 1: Find the dimensions of the box
First, we need to determine the dimensions of the box. Since 3-inch squares are cut out of each corner, the length and width of the resulting base will be reduced by 6 inches (3 inches on each side). The height of the box is not affected, so it remains as 3 inches.
Step 2: Calculate the volume of the box
Now, we can calculate the volume of the box using the formula: V = length × width × height.
Given that the volume of the box is 243 cubic inches and the height is 3 inches, we have:
243 = (length - 6) × (width - 6) × 3
Step 3: Solve for length and width
Simplify the equation:
81 = (length - 6) × (width - 6)
Since we know the length and width of a square are the same, let's substitute length = width = s:
81 = (s - 6) × (s - 6)
81 = (s - 6)^2
Step 4: Solve the equation
Taking the square root of both sides to eliminate the square:
√81 = √(s - 6)²
9 = s - 6
Now, isolate s:
s = 9 + 6
s = 15
Step 5: Determine the length of a side
Therefore, the length of a side of the original cardboard square is 15 inches.