Question

The area of the region bounded by the lines \( \frac{x}{7\sqrt{3a}} + \frac{y}{b} = 4 \), \( x = 0 \), and \( y = 0 \) is:

• A. \(56\sqrt{3ab}\)
• B. \( 56a \)
• C. \( \frac{ab}{2} \)
• D. \( 3ab \)

Hint:

When finding the area of a triangle with vertices on the coordinate axes, use the formula for the area of a right triangle: \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \). The base and height are the distances along the \( x \)-axis and \( y \)-axis, respectively, which can be found by determining the intercepts of the line with the axes. Simplify the expression carefully to find the correct area.

Answer 1

The Correct Option is A

The equation of the line is:

\[ \frac{x}{7\sqrt{3a}} + \frac{y}{b} = 4. \]

Rewriting:

\[ b \cdot x + 7\sqrt{3a} \cdot y = 28\sqrt{3ab}. \]

To find the intercepts:

1. When \( x = 0 \), \( y = 4b \).
2. When \( y = 0 \), \( x = 28\sqrt{3a} \).

The lines \( x = 0 \) and \( y = 0 \), together with \( \frac{x}{7\sqrt{3a}} + \frac{y}{b} = 4 \), form a triangle with vertices:

\[ (0, 0), \quad (28\sqrt{3a}, 0), \quad (0, 4b). \]

The area of the triangle is:

\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height}, \]

where the base is \( 28\sqrt{3a} \) and the height is \( 4b \):

\[ \text{Area} = \frac{1}{2} \times (28\sqrt{3a}) \times (4b). \]

Simplify:

\[ \text{Area} = \frac{1}{2} \times 112\sqrt{3ab} = 56\sqrt{3ab}. \]

Thus, the area of the region is:

\[ 56\sqrt{3ab}. \]

Answer 2

The equation of the line is:

\[ \frac{x}{7\sqrt{3a}} + \frac{y}{b} = 4. \]

Step 1: Rewrite the equation:

Multiply the entire equation by \( b \cdot 7\sqrt{3a} \) to eliminate the fractions: \[ b \cdot x + 7\sqrt{3a} \cdot y = 28\sqrt{3ab}. \]

Step 2: Find the intercepts:

To find the intercepts, we substitute \( x = 0 \) for the \( y \)-intercept and \( y = 0 \) for the \( x \)-intercept: - When \( x = 0 \), we have \( y = 4b \). - When \( y = 0 \), we have \( x = 28\sqrt{3a} \).

Step 3: Form the triangle:

The lines \( x = 0 \), \( y = 0 \), and the equation \( \frac{x}{7\sqrt{3a}} + \frac{y}{b} = 4 \) form a triangle with the vertices at: \[ (0, 0), \quad (28\sqrt{3a}, 0), \quad (0, 4b). \]

Step 4: Calculate the area of the triangle:

The area of a triangle is given by: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height}. \] Here, the base is \( 28\sqrt{3a} \) and the height is \( 4b \). Therefore, the area is: \[ \text{Area} = \frac{1}{2} \times (28\sqrt{3a}) \times (4b). \] Simplify: \[ \text{Area} = \frac{1}{2} \times 112\sqrt{3ab} = 56\sqrt{3ab}. \]

Conclusion: Thus, the area of the region is:

\[ 56\sqrt{3ab}. \]