- Identify the triangle's vertices as the origin \((0,0)\), the x-intercept \((8,0)\), and the y-intercept \((0,-6)\).
- Determine the lengths of the triangle's legs along the axes as \(8\) and \(6\).
- Apply the triangle area formula \(\frac{1}{2} \times \text{base} \times \text{height}\) to find the result of \(24 \, u^2\).
- Select option B, which matches the calculated area.
- \(\boxed{24 \, u^2}\)
Explanation
- Analyze the geometric configuration of the problem.
The problem asks for the area of the triangle formed by the line \(3x - 4y = 24\) and the coordinate axes. The triangle is bounded by the line itself, the x-axis, and the y-axis. The vertices of this triangle are the origin \((0, 0)\), the x-intercept of the line, and the y-intercept of the line.
- Calculate the intercepts and the lengths of the legs of the triangle.
To find the x-intercept, we set \(y = 0\) in the given linear equation \(3x - 4y = 24\):
\[3x - 4(0) = 24\]
\[3x = 24\]
\[x = 8\]
So, the x-intercept is at the point \((8, 0)\). The length of the base of the triangle along the x-axis is \(b = |8| = 8\).
To find the y-intercept, we set \(x = 0\) in the given linear equation \(3x - 4y = 24\):
\[3(0) - 4y = 24\]
\[-4y = 24\]
\[y = -6\]
So, the y-intercept is at the point \((0, -6)\). The height of the triangle along the y-axis is \(h = |-6| = 6\).
- Calculate the area of the triangle and identify the correct option.
The triangle formed is a right-angled triangle with vertices at \((0, 0)\), \((8, 0)\), and \((0, -6)\). The legs of the right triangle have lengths \(b = 8\) and \(h = 6\). The area \(A\) of a right-angled triangle is given by the formula:
\[A = \frac{1}{2} \cdot b \cdot h\]
Substituting the known values:
\[A = \frac{1}{2} \cdot 8 \cdot 6\]
\[A = \frac{1}{2} \cdot 48\]
\[A = 24\]
Thus, the area of the triangle is \(24 \, u^2\). This corresponds to option B.
Examples
In urban planning, architects often need to calculate the area of land bounded by straight paths or roads represented as linear equations on a coordinate grid. For instance, if two roads are designated by \(3x - 4y = 24\) and the city boundary is defined by the axes, the area of the corner lot can be calculated exactly as we did: \(A = \frac{1}{2} \times 8 \times 6 = \boxed{24 \, u^2}\). This helps in determining property taxes, construction costs, and zone planning.