如果矩形的长度减少 20%，那么宽度必须增加百分之几才能保持原来的面积？

数学是一门负责计算的学科。并且，根据要执行的计算或运算的类型，数学分为不同的分支，如代数、几何、算术等。

测量是处理各种形状的周长、面积、体积等参数计算的分支，无论是二维还是三维。

在二维形状中，对象由长度和宽度或可以在平面上表示的任何两个维度组成。而三维形状，对象被放置在现实世界中，并具有长度、宽度和高度三个维度。

2D 和 3D 形状的一些基本公式

2D 形状

• 长方形

Area = length × breadth

Perimeter = 2(length+breadth)

• 正方形

Area = (side)2

Perimeter = 4(side)

• 圆圈

Diameter = 2 × radius

Area = π × (radius)²

• 三角形

Area = 1/2 breadth × height

3D 形状

• 立方体

Volume = (side)³

Lateral surface area = 4 × (side)²

Total surface area = 6 × (side)²

• 长方体

Volume = length × breadth × height

Lateral surface area = 2 × height(l+b)

Total surface area = 2(lb+lh+hb)

• 领域

Volume = 4/3πr³

Surface area = 4πr²

• 锥体

Volume = 1/3πr²h

Total surface area = πr (l+radius)

如果矩形的长度减少 20%，那么宽度必须增加百分之几才能保持原来的面积？

解决方案：

Let x and y represent the length and breadth of the rectangle respectively.

As we know the original area of the rectangle by the standard formula 

Area of rectangle(A) = l × b

Area of given rectangle = xy

According to the question the length of rectangle is reduced by 20%. So, the new length would be

=>x-20/100x

=>x(1-1/5)

=>4/5x

And, k% be the increase in breadth to maintain original area.

=>y+k/100y

=>y(1+k/100)

As it is stated that the original and new area needs to be same.

=>Original area=new area

=>xy=(4/5)x(1 +k/100)y

=>1=(4/5)(100+k/100)

=>100+k/100 = 5/4

=>100+k = 125

=>k = 125-100

=>k = 25

Hence, the breadth needs to be increased by 25% to maintain original area.

示例问题

问题1。当矩形的长和宽增加 40% 时，面积会增加多少。

解决方案：

Let x and y be the length and breadth of the rectangle respectively.

The area of rectangle by standard formula will be

=> area of rectangle(A)= xy

According to the question,

Length of rectangle is increased by 40% = x+40/100x

=>x+40/100x

=>x(1+40/100)

=>140/100x

=>7/5x=1.4x

Breadth of rectangle is increased by 40%=y+40/100y

=>y+40/100y

=>y(1+40/100)

=>140/100y

=>7/5y=1.4y

Now, the new area of rectangle will be =1.4x x 1.4y =1.96xy

And, increase in area of rectangle =1.96xy-xy = 0.96xy

Increase in percentage of area of rectangle = 0.96xy/xy x 100%

                                                                    = 96%

问题2。当矩形的长度增加 20%，宽度增加 40% 时，面积会增加多少百分比。

解决方案：

Let x and y be the length and breadth of the rectangle respectively.

The area of rectangle by standard formula will be

=> area of rectangle(A)= xy

According to the question,

Length of rectangle is increased by 20%=x+20/100x

=>x+20/100x

=>x(1+20/100)

=>120/100x

=>6/5x=1.2x

Breadth of rectangle is increased by 40%=y+40/100y

=>y+40/100y

=>y(1+40/100)

=>140/100y

=>7/5y=1.4y

Now, the new area of rectangle will be =1.2x x 1.4y =1.68xy

And, increase in area of rectangle =1.68xy-xy = 0.68xy

Increase in percentage of area of rectangle=0.68xy/xy x 100%

                                                                      =68%

问题3。矩形的长度是其宽度的三倍。它的周长是40m。找出长度和宽度。

解决方案：

Let the breadth of the rectangle be x

As per the question the length is double the value of breadth so it will be 3x

perimeter(P) = 40cm

Now, by the formula,

Perimeter of rectangle(P) = 2(l+b)

=>40 = 2(x+3x)

=>40 = 2.4x

=>x=40/8

=>x=5cm

=>3x=3 . 5=15cm

Hence, the length and breadth of the rectangle are 15cm and 5cm respectively.