解答题 小明用橡皮泥捏了一个正方体，然后又拿出一些橡皮泥，沿这个正方体一个面的方向加长1厘米，得到一个长方体（如下图），这个长方体比原正方体的表面积增加了12平方厘米，求这个长方体的表面积和体积 【分析】根据题意可知，增加部分面积等于4个长等于正方体棱长，宽是1厘米的长方形面积和，用增加部分的面积÷4，求出一个长方形的面积；再根据长方形面积公式：面积＝长×宽；长＝面积÷宽，代入数据，求出正方体的棱长，进而求出增加后长方体的长、宽、高，再根据长方体表面积公式：面积＝（长×宽＋长×高＋宽×高）×2，长方体体积公式：体积＝长×宽×高，代入数据，即可解答。 【详解】正方体的棱长： 12÷4÷1 ＝3÷1 ＝3（厘米） 长方形的长：1＋3＝4（厘米），宽是3厘米，高是3厘米。 表面积：（4×3＋4×3＋3×3）×2 ＝（12＋12＋9）×2 ＝（24＋9）×2 ＝33×2 ＝66（平方厘米） 体积：4×3×3 ＝12×3 ＝36（立方厘米） 答：这个长方体的表面积66平方厘米，体积36立方厘米。 【点睛】本题考查长方体表面积公式、长方体体积公式的应用，关键明确增加部分的面积与正方体棱长之间的关系是解答本题的关键。---**Chart/Diagram Description:** * **Type:** This is a diagram illustrating a transformation or process involving a 3D geometric solid. * **Main Elements:** * **Left Object (Initial State):** A 3D representation of a cube. * **Faces:** The front face is white, and the top and right side faces are shaded light grey, indicating a perspective view. * **Edges:** All visible edges are represented by solid black lines. * **Arrow:** A large, hollow, right-pointing arrow is positioned between the left object and the right object, indicating a transition or transformation from the cube to the object on the right. * **Right Object (Final State):** A 3D representation of a rectangular prism (cuboid). * **Faces:** Similar to the cube, its front face is white, and its top and right side faces are shaded light grey. * **Dimensions:** The cuboid appears to be elongated horizontally compared to the initial cube. * **Internal Feature:** A vertical dashed line is drawn on the front face of the cuboid, extending upwards and continuing as a dashed line on the top face. This dashed line appears to divide the cuboid approximately in half along its length, suggesting a section, a cut, or a partition.