Question

Enrique is constructing a doorway. The width of the doorway is framed from an 8-foot board with a length of 2√3
feet cut from it. The length removed from this board is attached to another 8-foot board to frame the height of the door. What is the area of the opening of the doorway Enrique is building?(2 points)

Answer 1

To find the area of the opening of the doorway, we need to determine both the width and height of the doorway.

1. **Calculating the Width of the Doorway**:
- Enrique frames the width from an 8-foot board and cuts a length of \(2\sqrt{3}\) feet from it.
- The remaining width of the doorway is:
\[
8 \text{ feet} - 2\sqrt{3} \text{ feet}
\]

2. **Calculating the Height of the Doorway**:
- The length that was cut off, which is \(2\sqrt{3}\) feet, is attached to another 8-foot board to frame the height of the door.
- Thus, the height of the doorway is \(2\sqrt{3}\) feet.

3. **Calculating the Area of the Opening**:
- The area \(A\) of the doorway can be found using the formula:
\[
A = \text{width} \times \text{height}
\]
- Substituting in the expressions for width and height:
\[
A = (8 - 2\sqrt{3}) \times (2\sqrt{3})
\]

- Expanding this expression:
\[
A = (8 \times 2\sqrt{3}) - (2\sqrt{3} \times 2\sqrt{3})
\]
\[
A = 16\sqrt{3} - 4 \times 3
\]
\[
A = 16\sqrt{3} - 12
\]

Thus, the area of the opening of the doorway that Enrique is building is:
\[
A = 16\sqrt{3} - 12 \text{ square feet}
\]

This is the final expression for the area of the doorway. If you need a numerical value, you can approximate \(\sqrt{3} \approx 1.732\):
\[
A \approx 16 \times 1.732 - 12 \approx 27.712 - 12 \approx 15.712 \text{ square feet}
\]

However, the exact area is \(16\sqrt{3} - 12\) square feet.

Answer 2

I got 64 - 16 sqrt 3

Answer 3

Let's reevaluate the problem step by step.

1. **Width of the Doorway**:
- The original length of the board is 8 feet.
- A length of \(2\sqrt{3}\) feet is cut from this board:
\[
\text{Width} = 8 - 2\sqrt{3}
\]

2. **Height of the Doorway**:
- The height of the doorway is determined from the cut piece, which is attached to another 8-foot board. This piece has a length of \(2\sqrt{3}\) feet:
\[
\text{Height} = 2\sqrt{3}
\]

3. **Area of the Doorway**:
- The area \(A\) of the doorway is given by:
\[
A = \text{Width} \times \text{Height}
\]
- Plugging in the values we have:
\[
A = (8 - 2\sqrt{3}) \times (2\sqrt{3})
\]

- Expanding this:
\[
A = 8 \times 2\sqrt{3} - 2\sqrt{3} \times 2\sqrt{3}
\]
\[
A = 16\sqrt{3} - (2\sqrt{3})^2
\]
\[
A = 16\sqrt{3} - 4 \times 3
\]
\[
A = 16\sqrt{3} - 12
\]

From the process above, the area of the doorway is \(16\sqrt{3} - 12\) square feet, which matches my previous calculation.

Let's verify what you calculated \(64 - 16\sqrt{3}\):
- Your area seems calculated differently. Let's break down your calculation to see if there’s a misunderstanding.
- The only way to be \(64 - 16\sqrt{3}\) would be if you took an entirely different interpretation or if numbers were mixed up.

From the calculations provided, the area of the doorway Enrique is building remains \(16\sqrt{3} - 12\) square feet. If there’s something specific you'd like to discuss about how to achieve \(64 - 16\sqrt{3}\), please let me know!