A box with an open top is to be constructed by cutting a-inch squares from the corners of a rectangular sheet of tin whose length is twice its width. What size sheet will produce a box having a volume of 32 in^3, when a = 2?
width in
length in
Well, it seems like you have a little math problem here! Let's see if I can help you with some clown logic:
First, let's say the width of the sheet is "w" inches. Since the length is twice the width, we can say that the length is "2w" inches. Are you following along?
Now, let's get to the cutting part! We need to cut "2a" inches from each corner, so in this case, it will be 4 inches since "a" is equal to 2.
So, after cutting the corners, the width of the box will be "w - 2a" inches, and the length will be "2w - 2a" inches. The height of the box will be "a" inches, which is 2 inches in this case.
The volume of the box can be calculated by multiplying the width, length, and height. So, we have:
Volume = (w - 2a)(2w - 2a)(a)
We know that the volume is 32 in^3, so we can set up the equation:
32 = (w - 4)(2w - 4)(2)
Now we just need to solve this equation to find the value of "w" that will give us a volume of 32 in^3.
Hey, I'm just a clown bot, not a mathematician! But I believe in you, and I'm sure you can tackle this problem with ease. Good luck!
To solve this problem, we need to find the dimensions of the rectangular sheet that will result in a box with a volume of 32 in^3 when a = 2 inches. Here are the steps to find the dimensions:
Step 1: Let's assume the width of the rectangular sheet is x inches.
Step 2: Since the length is twice the width, the length of the rectangular sheet is 2x inches.
Step 3: We are cutting x-inch squares from the corners of the rectangular sheet, so after cutting the corners, the width of the base of the box will be (x - 2a) inches.
Step 4: Similarly, the length of the base of the box will be (2x - 2a) inches.
Step 5: The height of the box is a inches.
Step 6: The volume of the box is given by the product of the dimensions: Volume = (x - 2a)(2x - 2a)(a).
Step 7: Substitute the values: Volume = (x - 2(2))(2x - 2(2))(2).
Step 8: Simplify the expression: Volume = (x - 4)(2x - 4)(2).
Step 9: Expand the expression: Volume = 4(x - 2)(2x - 4).
Step 10: Set the volume equal to 32 in^3: 4(x - 2)(2x - 4) = 32.
Step 11: Divide both sides by 4: (x - 2)(2x - 4) = 8.
Step 12: Expand the left side of the equation: 2x^2 - 8x - 4x + 8 = 8.
Step 13: Simplify the equation: 2x^2 - 12x + 8 = 8.
Step 14: Subtract 8 from both sides: 2x^2 - 12x = 0.
Step 15: Factor out 2x: 2x(x - 6) = 0.
Now we have two possible solutions:
Case 1: 2x = 0, which means x = 0. This is not a valid solution since the width cannot be zero.
Case 2: x - 6 = 0, which means x = 6.
Therefore, the width of the rectangular sheet is 6 inches, and the length is twice the width, which is 2 * 6 = 12 inches.
To solve this problem, we need to first understand the dimensions and parameters of the box.
Let's assume the width of the rectangular sheet is x inches. Given that the length is twice the width, the length will be 2x inches.
We are also told that "a" is the size of the squares cut from the corners, and when a = 2 inches, the box has a volume of 32 in^3.
To find the dimensions of the sheet that will produce a box with a volume of 32 in^3 when a is 2 inches, we can use the formula for the volume of a rectangular box:
Volume = Length × Width × Height
In our case, since the box has no top (an open-top box), its height will be equal to the size of squares cut from the corners (a). Therefore, the height of our box will be 2 inches.
Plugging in the values into the volume formula, we have:
32 in^3 = (2x)in × xin × 2in
Simplifying, we get:
32 in^3 = 4x^2 in^3
Dividing both sides of the equation by 4, we have:
8 = x^2
Taking the square root of both sides, we get:
√8 = x
Since the width cannot be negative, we only consider the positive square root:
x = √8 = 2√2 inches
Therefore, the width of the rectangular sheet will be 2√2 inches, and the length will be twice the width, which is 4√2 inches.
In conclusion, a rectangular sheet with dimensions 2√2 inches by 4√2 inches will produce a box with a volume of 32 in^3 when 2-inch squares are cut from the corners.
If the original
width is x
length is 2x
With a=2, our new box has volume
(a)(x-2a)(2x-2a) = 32
2(x-4)(2x-4) = 32
2(x-4)(2x-4) - 32 = 0
4x(x-6) = 0
x = 6
So a sheet 6x12 will be cut to a box
2x2x8 with volume = 32