求阴影周长和面积

Math

看图，做题目。

题目分析

题目描述了一个大长方形中剪下两个大小相同的小长方形，留下一个“T”形的阴影部分。我们需要用含 x,yx,y 的代数式表示阴影部分的周长和面积，并在给定 x=2,y=2.5x=2,y=2.5 时计算阴影部分的面积。

解答(1) 阴影部分的周长

阴影部分的周长是围绕“T”形图形的所有边的长度之和。假设每个小长方形的宽度为 xx，高度为 yy，则阴影部分的周长可以表示为：

周长=2x+4y周长=2x+4y

(2) 阴影部分的面积

阴影部分的面积是大长方形的面积减去两个小长方形的面积。假设大长方形的宽度为 2x2x，高度为 2y2y，则阴影部分的面积可以表示为：

面积=(2x×2y)−2(x×y)=4xy−2xy=2xy面积=(2x×2y)−2(x×y)=4xy−2xy=2xy

(3) 当 [size=1.21em]x=2,y=2.5x=2,y=2.5 时，计算阴影部分的面积

将 x=2x=2 和 y=2.5y=2.5 代入面积公式：

面积=2xy=2×2×2.5=10面积=2xy=2×2×2.5=10

最终答案

• 阴影部分的周长为 2x+4y2x+4y。
• 阴影部分的面积为 2xy2xy。
• 当 x=2,y=2.5x=2,y=2.5 时，阴影部分的面积为 10。
  
  

Problem Analysis

The problem describes a large rectangle with two smaller rectangles of the same size cut out, leaving a "T"-shaped shaded area. We need to express the perimeter and area of the shaded part using algebra with xx and yy, and then calculate the area when x=2x=2 and y=2.5y=2.5.

Solution(1) Perimeter of the Shaded Area

The perimeter is the total length around the "T"-shaped figure. If each small rectangle has a width of xx and a height of yy, the perimeter of the shaded area can be expressed as:

Perimeter=2x+4yPerimeter=2x+4y

(2) Area of the Shaded Area

The area of the shaded part is the area of the large rectangle minus the areas of the two small rectangles. If the large rectangle has a width of 2x2x and a height of 2y2y, the area of the shaded part is:

Area=(2x×2y)−2(x×y)=4xy−2xy=2xyArea=(2x×2y)−2(x×y)=4xy−2xy=2xy

(3) Calculating the Area When [size=1.21em]x=2x=2 and [size=1.21em]y=2.5y=2.5

Substitute x=2x=2 and y=2.5y=2.5 into the area formula:

Area=2xy=2×2×2.5=10Area=2xy=2×2×2.5=10

Final Answers

• The perimeter of the shaded area is 2x+4y2x+4y.
• The area of the shaded area is 2xy2xy.
• When x=2x=2 and y=2.5y=2.5, the area of the shaded part is 10.