Area enclosed by equation `y = |x| - 5` with x-axis is

Options

A. 25 sq. unit
B. `12.5` sq. unit
C. 50 sq. unit
D. None of these

Answer

To find the area enclosed by the equation \( y = |x| - 5 \) with respect to the x-axis, we will follow these steps:

### Step 1: Understand the Equation
The equation \( y = |x| - 5 \) represents a V-shaped graph that opens upwards. The vertex of this graph is at the point where \( y = 0 \).

### Step 2: Find the x-intercepts
To find the x-intercepts, set \( y = 0 \):
\[ 0 = |x| - 5 \]
This implies:
\[ |x| = 5 \]
Thus, \( x = 5 \) and \( x = -5 \).

### Step 3: Identify the Points of Interest
The points of interest are:
- \( (5, 0) \)
- \( (-5, 0) \)
- The vertex point where \( x = 0 \):
\[ y = |0| - 5 = -5 \quad \text{(Point: (0, -5))} \]

### Step 4: Plot the Points
Plot the points on a graph:
- Point A: \( (-5, 0) \)
- Point B: \( (0, -5) \)
- Point C: \( (5, 0) \)

### Step 5: Determine the Shape
The points form a triangle with the base along the x-axis from \( (-5, 0) \) to \( (5, 0) \) and the vertex at \( (0, -5) \).

### Step 6: Calculate the Base and Height of the Triangle
- The base \( AC \) is the distance between \( (-5, 0) \) and \( (5, 0) \):
\[ \text{Base} = 5 - (-5) = 10 \text{ units} \]
- The height \( OB \) is the vertical distance from the x-axis to the vertex \( (0, -5) \):
\[ \text{Height} = 5 \text{ units} \]

### Step 7: Calculate the Area of the Triangle
The area \( A \) of a triangle is given by the formula:
\[ A = \frac{1}{2} \times \text{Base} \times \text{Height} \]
Substituting the values:
\[ A = \frac{1}{2} \times 10 \times 5 = 25 \text{ square units} \]

### Conclusion
The area enclosed by the equation \( y = |x| - 5 \) with respect to the x-axis is \( 25 \) square units.

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