题目 16
文字信息：
题干：
Jim has a triangular shelf system that attaches to his showerhead. The total height of the system is 18 inches, and there are three parallel shelves as shown above. What is the maximum height, in inches, of a shampoo bottle that can stand upright on the middle shelf?
其他相关文字：
shelf (标签，指向搁板)
shampoo (标签，指向洗发水瓶)
18 in (标注，指示整个系统的总高度)
2x, 3x, x (标注，指示三角形斜边被分割成的三段的相对长度)
图表/示意图描述：
类型：几何示意图。
主要元素：
一个大的等腰三角形（隐含，因为搁板是水平的）。
该大三角形内部有两条水平线，将大三角形分割成一个小三角形（顶部）和两个梯形（中间和底部），形成了三个置物空间。这三条水平线（包括大三角形的底边）构成了题目所说的 "three parallel shelves"。
点与线：
大三角形的顶点在最上方。
从顶点沿左侧斜边向下，依次被标记为长度 2x, 3x, x 的三段。
两条水平的搁板线与大三角形的底边平行。
洗发水瓶（shampoo）被画成一个矩形，放置在中间的搁板上（即从下往上数第二层搁板，或者说是由斜边长度为 3x 的部分所对应的梯形区域的下底面）。
形状：
整体为一个大三角形。
内部包含一个洗发水瓶，示意为矩形。
搁板为水平直线段。
标签与注释：
"shelf" 标签箭头指向水平的搁板线。
"shampoo" 标签箭头指向洗发水瓶。
右侧一个垂直的双箭头线段标注了整个三角形系统的高度为 "18 in"。
左侧斜边从上到下被分为三段，分别标注为 "2x", "3x", "x"。
相对位置与方向：
洗发水瓶竖直放置在中间的搁板上，位于由斜边3x定义的区域内。
搁板是水平且互相平行的。
整个系统总高度为18英寸。
视频信息
答案文本
视频字幕
Jim has a triangular shelf system in his shower with a total height of 18 inches. The system has three parallel shelves. The left side of the triangle is divided into three segments with relative lengths of 2x, 3x, and x from top to bottom. We need to find the maximum height of a shampoo bottle that can stand upright on the middle shelf.
To solve this problem, we need to find the heights of the shelves. The total side length ratio is 2x plus 3x plus x, which equals 6x. The top shelf is positioned at a ratio of 2x divided by 6x, which equals one-third of the total height. So the top shelf is at 6 inches from the top. The middle shelf is positioned at a ratio of 5x divided by 6x, which is five-sixths of the total height. So the middle shelf is at 15 inches from the top. The maximum height of a bottle that can stand on the middle shelf is the distance between the middle and top shelves, which is 15 minus 6, or 9 inches.
Let's verify our solution. The total height of the triangular shelf system is 18 inches. The left side is divided in the ratio 2x to 3x to x from top to bottom. The top shelf is positioned at a height of 6 inches from the top, which is one-third of the total height. The middle shelf is positioned at a height of 15 inches from the top, which is five-sixths of the total height. Therefore, the maximum height of a shampoo bottle that can stand upright on the middle shelf is 15 minus 6, which equals 9 inches. This is our final answer.
To summarize what we've learned: The triangular shelf system has a total height of 18 inches with three parallel shelves. The left side is divided in the ratio 2x to 3x to x, giving a total ratio of 6x. The top shelf is positioned at 6 inches from the top, which is one-third of the total height. The middle shelf is positioned at 15 inches from the top, which is five-sixths of the total height. Therefore, the maximum height of a shampoo bottle that can stand upright on the middle shelf is 9 inches, which is the distance between the top and middle shelves.