Suppose she wants to carpet ifer bedroom and add basehoards. From which flooting store should she order the materials to get the better deal? Explain your reasoning: Part D Carmen decided that she also needs space for a new desk. She wants to place: the desk between the two trangular shelves with one corner at the point $$ (-10,3) $$ The desk is 3 teet long and $$ 1 \frac{1}{2} $$ feet wide. What are possible coordinates lor the other three corners of the desk? What is the area of the space the desk will cover?

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To solve the problem, we need to determine the possible coordinates for the other three corners of the desk and calculate the area it will cover.

### Step 1: Understanding the Desk's Position
Carmen wants to place the desk with one corner at the point $$(-10, 3)$$. The desk is 3 feet long and $$1 \frac{1}{2}$$ feet wide, which can be converted to an improper fraction:
$$
1 \frac{1}{2} = \frac{3}{2} \text{ feet}
$$

### Step 2: Possible Configurations
The desk can be oriented in different ways. We will consider two main orientations:
1. **Horizontal Orientation**: The length (3 feet) is along the x-axis.
2. **Vertical Orientation**: The width ($$\frac{3}{2}$$ feet) is along the x-axis.

#### Case 1: Horizontal Orientation
- The coordinates of the corners can be calculated as follows:
  - Corner 1: $$(-10, 3)$$ (given)
  - Corner 2: $$(-10 + 3, 3) = (-7, 3)$$ (3 feet to the right)
  - Corner 3: $$(-10, 3 - \frac{3}{2}) = (-10, 1.5)$$ (1.5 feet down)
  - Corner 4: $$(-7, 3 - \frac{3}{2}) = (-7, 1.5)$$ (combination of the above)

So, the coordinates for the corners in this orientation are:
- $$(-10, 3)$$
- $$(-7, 3)$$
- $$(-10, 1.5)$$
- $$(-7, 1.5)$$

#### Case 2: Vertical Orientation
- The coordinates of the corners can be calculated as follows:
  - Corner 1: $$(-10, 3)$$ (given)
  - Corner 2: $$(-10, 3 + 3) = (-10, 6)$$ (3 feet up)
  - Corner 3: $$(-10 - \frac{3}{2}, 3) = (-11.5, 3)$$ (1.5 feet to the left)
  - Corner 4: $$(-10 - \frac{3}{2}, 6) = (-11.5, 6)$$ (combination of the above)

So, the coordinates for the corners in this orientation are:
- $$(-10, 3)$$
- $$(-10, 6)$$
- $$(-11.5, 3)$$
- $$(-11.5, 6)$$

### Step 3: Area Calculation
The area $$A$$ of the rectangle (desk) can be calculated using the formula:
$$
A = \text{length} \times \text{width}
$$
Substituting the values:
$$
A = 3 \times \frac{3}{2} = \frac{9}{2} \text{ square feet} = 4.5 \text{ square feet}
$$

### Summary of Results
- Possible coordinates for the corners of the desk:
  - **Horizontal Orientation**: 
    - $$(-10, 3)$$, $$(-7, 3)$$, $$(-10, 1.5)$$, $$(-7, 1.5)$$
  - **Vertical Orientation**: 
    - $$(-10, 3)$$, $$(-10, 6)$$, $$(-11.5, 3)$$, $$(-11.5, 6)$$
- Area of the space the desk will cover: 
  - $$4.5$$ square feet.
Possible desk corner coordinates: - Horizontal: $$(-10, 3)$$, $$(-7, 3)$$, $$(-10, 1.5)$$, $$(-7, 1.5)$$ - Vertical: $$(-10, 3)$$, $$(-10, 6)$$, $$(-11.5, 3)$$, $$(-11.5, 6)$$ Area of the desk space: 4.5 square feet

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