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.