Question

图形 $|3x| + |4y| = 12$ 围成的面积是多少？

What is the area enclosed by the graph of $|3x| + |4y| = 12?$

$6$

$12$

$16$

$24$

$25$

令 $y = 0$ ，得 $|3x| = 12$ ，所以 $x = \pm 4$ 。令 $x = 0$ ，得 $|4y| = 12$ ，所以 $y = \pm 3$ 。

该图形是一个菱形，顶点为 $(\pm 4, 0)$ 和 $(0, \pm 3)$ ，所以两条对角线的长度分别为 $8$ 和 $6$ 。

它的面积为 $\dfrac12 \cdot 8 \cdot 6 = 24$ 。

所以正确答案是 D。

Setting $y = 0$ gives $|3x| = 12,$ so $x = \pm 4.$ Setting $x = 0$ gives $|4y| = 12,$ so $y = \pm 3.$

The graph is a rhombus with vertices $(\pm 4, 0)$ and $(0, \pm 3),$ so its diagonals have lengths $8$ and $6.$

Its area is $\dfrac12 \cdot 8 \cdot 6 = 24.$

Thus, the correct answer is D.