如果长方体的长度加倍，高度相同，宽度减半，长方体的体积会发生什么变化？

长方体是一个三维立体物体。它也被称为正六面体，是五个柏拉图立体之一。所有的边至少共享一条公共边。一个长方体的结构可以这样定义：每个面都连接到四个顶点和四个边，顶点连接到三个边和三个面，并且边连接到两个面和两个顶点。在长方体的情况下，长度、宽度或高度可能相等，也可能不相等。

长方体的性质

• 所有的面本质上都是矩形的。
• 所有的角度都是直角
• 长方体的相对面相等。

长方体的体积

长方体的体积等于图形中占据的空间量。任何三维图形的体积取决于三个边的长度，即它的长、宽和高。它可以被认为是一个实心矩形。让我们假设长方体的高度为 h，l 其长度和宽度分别用 b 个单位表示。

除此之外，让我们假设 V 是长方体的体积。推导出其公式，

长方体的体积=底面积×高

长方体的底面积 = l × b

Volume of a cuboid, V = l × b × h = lbh

还，

长方体的体积=长方体的面积×高

如果长方体的长度加倍，高度相同，宽度减半，长方体的体积会发生什么变化？

解决方案：

Let us assume l to be the length of cuboid and b to be its breadth. Also, let us assume h to be the height of the cuboid. 

So, 

Volume of a cuboid = l × b × h 

Now,  

Length of a cuboid becomes = 2l 

Breadth = b/2 

Height = h 

Volume of cuboid , on simplifying becomes, 

2l × b/2 × h 

= l × b × h 

Therefore, the final volume of the cuboid remains same. 

示例问题

问题 1. 如果一个长方体的体积是 24000 cm ³ 。那么如果长度增加一倍，高度相同，宽度减少一半，体积的变化呢？

解决方案：

Here we have to find the volume change in the cuboid if the dimensions are changed

Volume of cuboid = l × b × h = 24000 cm³

Changed dimensions

l = 2l

b = $\frac{b}{2}$

h = h

Therefore,

Changed dimensions volume = 2l × $\frac{b}{2}$ × h

Further simplifying 

Changed dimensions volume = l × b × h

Thus, we can clearly see that their is no change in the dimensions

As the dimensions are same 

Therefore,

Volume of cuboid is also same that is 24000 cm³.

问题2.如果长方体的体积是3000 m ³ ，长20 m 宽15 m，求长方体的高？

解决方案：

Here we have to find the height of the cuboid using its volume and 2 dimensions

As we know that,

Volume of cuboid = l × b × h 

According to the question

Volume of cuboid = 3000 m³

Substituting value of volume of cuboid

3000 =  l × b × h 

3000 = 20 × 15 × h

h = $\frac{3000}{20× 15}$

h = 10 m

Therefore,

Height of the cuboid is 10 m.

问题 3. 考虑一个长方体的长是宽的两倍，长方体的宽是高的两倍，长方体的体积是 8000 m ³ 。找到长方体的所有尺寸？

解决方案：

Here we have to find the dimensions of the cuboid by using the given volume 8000 m³.

Assume that the height of the cuboid is ‘x’.

According to the question,

Breadth of cuboid is double its height,

So, Breadth = 2 × height = 2 × x = 2x

And, Length of cuboid is double its breadth,

So, Length = 2 × breadth = 2 × 2x = 4x

Therefore,

Length = 4x, 

Breadth = 2x

Height = x

As we know that,

Volume of cuboid = l × b × h 

Volume of cuboid = 4x × 2x × x

8000 = 8x³

x³ = $\frac{8000}{8}$

x³ = 1000

x = $\sqrt[3]{1000}$

x = 10 m

Hence,

Height of the cuboid = x = 10 m

Breadth of the cuboid = 2x = 2 × 10 = 20 m

Length of the cuboid = 4x = 4 × 10 = 40 m

问题4.假设长方体的宽度减半，那么求长方体的体积变化了多少？

解决方案：

Here we have to find the change in volume of the cuboid if its breadth is halved

As we know that

Volume of cuboid = length × breadth × height 

According to the question

The breadth is halved

So, b = $\frac{b}{2}$

Therefore,

Volume of cuboid = l × $\frac{b}{2}$ × h 

Volume of cuboid = $\frac{l\times b\times h}{2}$

Volume of cuboid = $\frac{Volume\ of\ cuboid}{2}$. 

Hence, 

We can see that the volume of the cuboid gets halved if its breadth gets halved.

问题 5.^计算长 50m 宽 40m 高 10m 的水箱中能装满多少米 3 的水？

解决方案：

Here we have to find the amount of water that can be filled in a water tank of dimensions,

Length = 50 m

Breadth = 40 m

Height = 10 m

As we know that

Volume of cuboid = length × breadth × height 

Volume of cuboid = 50 × 40 × 10

Volume of cuboid = 20000 m³

Therefore,

20000 m³ water can be filled in the water tank.