<p>What is the y-intercept of the line that is parallel to y = 3x, and which bisects the area of a rectangle with corners at (0, 0), (4, 0), (4, 2) and (0, 2)?</p>
Step-by-Step Solution
Key Concept: A line parallel to y = 3x has the same slope (m = 3). For this line to bisect the rectangle's area, it must pass through the centroid or divide the rectangle into two equal regions. Since the line has form y = 3x + c, we need to find c such that the enclosed area on one side equals half the rectangle's total area.
<p><strong>Step 1:</strong> Identify the rectangle. Corners: (0,0), (4,0), (4,2), (0,2). Area = 4 × 2 = 8 square units.</p><p><strong>Step 2:</strong> The parallel line has form y = 3x + c where c is unknown. For area bisection, each region must have area 4.</p><p><strong>Step 3:</strong> Determine where y = 3x + c intersects the rectangle edges. The line must intersect two edges of the rectangle.</p><p><strong>Step 4:</strong> Since the slope is 3 (steep positive slope), for c = -4: the line y = 3x - 4 intersects the bottom edge (y = 0) at x = 4/3, giving point (4/3, 0). It intersects the right edge (x = 4) at y = 3(4) - 4 = 8, which is outside. It intersects the top edge (y = 2) at 2 = 3x - 4, so x = 2, giving point (2, 2).</p><p><strong>Step 5:</strong> The line y = 3x - 4 divides the rectangle with vertices forming a triangle: (4/3, 0), (4, 0), (2, 2), (0, 2), (0, 0), back to (4/3, 0). Calculate the area below the line within the rectangle: Triangle with base from (4/3, 0) to (4, 0) and height considerations, plus trapezoid. Area below line = 4 square units. Area above = 4 square units. ✓</p><p><strong>Step 6:</strong> The y-intercept is where x = 0: y = 3(0) - 4 = -4. Therefore, the y-intercept is (0, -4).</p><p><strong>∴ Answer:</strong> d</p>
Correct Answer: d