What is the area (in square units) of the figure enclosed by the graphs of the lines x = −4, x = 2,y = −2 and y = 3?

Options

A. 12
B. 24
C. 15
D. 30

Answer

To find the area enclosed by the lines \( x = -4 \), \( x = 2 \), \( y = -2 \), and \( y = 3 \), we can follow these steps:

### Step 1: Understand the Lines
The lines given are:
- \( x = -4 \): a vertical line at \( x = -4 \)
- \( x = 2 \): a vertical line at \( x = 2 \)
- \( y = -2 \): a horizontal line at \( y = -2 \)
- \( y = 3 \): a horizontal line at \( y = 3 \)

### Step 2: Identify the Vertices of the Enclosed Figure
The area enclosed by these lines forms a rectangle. The vertices of the rectangle can be found by determining the points where these lines intersect:
- The intersection of \( x = -4 \) and \( y = -2 \) gives the point \( (-4, -2) \).
- The intersection of \( x = -4 \) and \( y = 3 \) gives the point \( (-4, 3) \).
- The intersection of \( x = 2 \) and \( y = -2 \) gives the point \( (2, -2) \).
- The intersection of \( x = 2 \) and \( y = 3 \) gives the point \( (2, 3) \).

### Step 3: Calculate the Lengths of the Sides
Now we can calculate the lengths of the sides of the rectangle:
- The width (horizontal distance) between the lines \( x = -4 \) and \( x = 2 \) is:
\[ \text{Width} = 2 - (-4) = 2 + 4 = 6 \]
- The height (vertical distance) between the lines \( y = -2 \) and \( y = 3 \) is:
\[ \text{Height} = 3 - (-2) = 3 + 2 = 5 \]

### Step 4: Calculate the Area
The area \( A \) of the rectangle can be calculated using the formula:
\[ A = \text{Width} \times \text{Height} \]
Substituting the values we found:
\[ A = 6 \times 5 = 30 \text{ square units} \]

### Final Answer
Thus, the area enclosed by the lines is \( 30 \) square units.

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