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Answer :
Sure! Let's go through the steps to find the correct equation that indicates the volume of the box is 76 cubic inches.
1. Understand the problem:
- We start with a rectangular piece of cardboard.
- Squares with sides of 3 inches are cut from each corner.
- The flaps are folded up to form an open box.
2. Define the dimensions:
- Let [tex]\( x \)[/tex] be the width of the original piece of cardboard.
- The length will then be [tex]\( x + 8 \)[/tex] since the cardboard is stated to be 8 inches longer than it is wide.
3. Adjust dimensions after cutting:
- When squares of 3 inches are cut from each corner, the length and width of the resulting base are reduced by 6 inches (3 inches cut from each side).
- New width [tex]\( = x - 6 \)[/tex]
- New length [tex]\( = x + 8 - 6 = x + 2 \)[/tex]
4. Dimensions of the open box:
- The height of the box is the side length of the cut squares, which is 3 inches.
- New width of base [tex]\( = x - 6 \)[/tex]
- New length of base [tex]\( = x + 2 \)[/tex]
- Height [tex]\( = 3 \)[/tex]
5. Set up the volume equation:
- Volume of a box [tex]\( = \text{length} \times \text{width} \times \text{height} \)[/tex]
- Substituting in our dimensions, the volume [tex]\( V \)[/tex] is given by:
[tex]\[
V = (x + 2) \times (x - 6) \times 3
\][/tex]
- We're given that the volume of the box is 76 cubic inches, so:
[tex]\[
(x + 2) \times (x - 6) \times 3 = 76
\][/tex]
6. Find the correct equation:
- This matches with option D:
[tex]\[
(x + 2)(x - 6)(3) = 76
\][/tex]
So, the correct equation that indicates the volume of the box is 76 cubic inches is:
D. [tex]\((x + 2)(x - 6)(3) = 76\)[/tex]
1. Understand the problem:
- We start with a rectangular piece of cardboard.
- Squares with sides of 3 inches are cut from each corner.
- The flaps are folded up to form an open box.
2. Define the dimensions:
- Let [tex]\( x \)[/tex] be the width of the original piece of cardboard.
- The length will then be [tex]\( x + 8 \)[/tex] since the cardboard is stated to be 8 inches longer than it is wide.
3. Adjust dimensions after cutting:
- When squares of 3 inches are cut from each corner, the length and width of the resulting base are reduced by 6 inches (3 inches cut from each side).
- New width [tex]\( = x - 6 \)[/tex]
- New length [tex]\( = x + 8 - 6 = x + 2 \)[/tex]
4. Dimensions of the open box:
- The height of the box is the side length of the cut squares, which is 3 inches.
- New width of base [tex]\( = x - 6 \)[/tex]
- New length of base [tex]\( = x + 2 \)[/tex]
- Height [tex]\( = 3 \)[/tex]
5. Set up the volume equation:
- Volume of a box [tex]\( = \text{length} \times \text{width} \times \text{height} \)[/tex]
- Substituting in our dimensions, the volume [tex]\( V \)[/tex] is given by:
[tex]\[
V = (x + 2) \times (x - 6) \times 3
\][/tex]
- We're given that the volume of the box is 76 cubic inches, so:
[tex]\[
(x + 2) \times (x - 6) \times 3 = 76
\][/tex]
6. Find the correct equation:
- This matches with option D:
[tex]\[
(x + 2)(x - 6)(3) = 76
\][/tex]
So, the correct equation that indicates the volume of the box is 76 cubic inches is:
D. [tex]\((x + 2)(x - 6)(3) = 76\)[/tex]
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