Course: 8.3 锥体量-interactive

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锥体量
Cones
::锥体

Aside from providing a place to put ice cream, cones play an important role in the construction of our society. In architecture and engineering, t riangles are the most important shape for building sturdy structures because they distribute pressure equally. Cones have this same property among 3-dimensional shapes. They are also important in aeronautics, the science or practice of travel through air, because they are able to move through air and liquid more efficiently than any other 3-dimensional shape. This ability to assist in efficient flight is why they are so commonly used in the design of planes and rocket ships. You will see an example of this later in the lesson .
::除了提供放置冰淇淋的地方外,锥体还在我们社会的建设中发挥重要作用。在建筑和工程中,三角体是建造坚固结构的最重要形状,因为它们平分压力。锥体在三维形状中具有同样的属性。在航空、科学或空中旅行实践方面,它们也很重要,因为它们能够比其他任何三维形状更高效地通过空气和液体流动。这种协助高效飞行的能力是它们为什么在飞机和火箭飞船的设计中经常被使用的原因。你将在以后的教训中看到这方面的一个例子。

Volume of Cones
::锥体量

T he question that defines is: how many cubes can fit inside a given shape ? However, there is no easy way to physically stack cubes inside a cone, so you need to be more clever. Luckily, cones have a relationship with cylinders which you can use to find the volume of a cone.
::问题的定义是: 有多少立方体可以安装在给定形状中? 然而, 在锥体内部没有简单的物理堆叠方式, 所以您需要更聪明。幸运的是, 锥体与圆柱有关系, 您可以用来找到锥体的体积。

Use the interactive below to explore this relationship.
::使用下面的交互方式来探索这种关系。

Using the Formula
::使用公式

Now that you know the volume of the cone will always be equal to one-third the volume of a cylinder with an equal radius and height , you can use this to write the formula to find the volume of a cone. Use the formula for the volume of a cylinder and divide by 3 .
::既然你知道锥体的体积总是等于圆锥体体体积的三分之一, 其半径和高度等于圆锥体体体积的三分之一, 您可以用这个写公式来找到圆锥体体体积。使用圆锥体体体体积的公式来计算圆锥体体体体积, 然后除以 3 。

$\begin{aligned} Volume = π \cdot {radius}^{2} \cdot height \div 3 \end{aligned}$
$\begin{align*}\text{Volume} = \pi \cdot \text{radius}^{2} \cdot \text{height} ÷ 3\end{align*}$
::卷号2QQQQQQQQQQQQQQQQ 3

This can also be written with a fraction:
::也可以用一个分数写入 :

$\begin{aligned} Volume = \frac{π \cdot {radius}^{2} \cdot height}{3} \end{aligned}$
$\begin{align*}\text{Volume} = \frac{\pi \cdot \text{radius}^{2} \cdot \text{height}}{3}\end{align*}$
::卷积2QQ 高度3

Example
::示例示例示例示例

The interactive below shows the dimensions , in feet, for a spire on top of a house. Find the volume of the spire. Use 3.14 as the value for $\begin{align*}\pi.\end{align*}$
::下面交互显示房屋顶部的圆柱形尺寸。查找圆柱形的体积。使用 3. 14 作为的值。

U se the same steps that you used to find the volume of a cylinder.
::使用与您用来找到圆柱体体积的步数相同的步数。

1. Write the formula.
::1. 编写公式。

$\begin{aligned} Volume = \frac{π \cdot {radius}^{2} \cdot height}{3} \end{aligned}$
$\begin{align*}\text{Volume} = \frac{\pi \cdot \text{radius}^{2} \cdot \text{height}}{3}\end{align*}$
::卷积2QQ 高度3

2. Substitute the dimensions of the shape into the formula. Use 3.14 as the value for $\begin{align*}\pi.\end{align*}$
::2. 将形状的尺寸替换为公式,使用3.14作为______的值。

$\begin{aligned} Volume = \frac{3.14 \cdot 4^{2} \cdot 6}{3} \end{aligned}$
$\begin{align*}\text{Volume} = \frac{3.14 \cdot {4}^{2} \cdot 6}{3}\end{align*}$
::卷号=3.144263

3. Use order of operations to simplify the formula.
::3. 使用操作顺序简化公式。

$\begin{aligned} Volume & = \frac{3.14 \cdot 4^{2} \cdot 6}{3} \\ Volume & = \frac{3.14 \cdot 16 \cdot 6}{3} \\ Volume & = \frac{3.14 \cdot 96}{3} \\ Volume & = \frac{301.44}{3} \end{aligned}$
$\begin{align*}\text{Volume} &= \frac{3.14 \cdot {4}^{2} \cdot 6}{3} \\ \ \\ \text{Volume} &= \frac{3.14 \cdot 16 \cdot 6}{3} \\ \ \\ \text{Volume} &= \frac{3.14 \cdot 96}{3} \\ \ \\ \text{Volume} &= \frac{301.44}{3}\end{align*}$
::卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷卷

The volume of the spire is 100.48 ft ³ .
::螺旋的体积为100.48平方英尺。

Discussion Question
::讨论问题

Are there any tricks you could have used to simplify the formula?
::你本可以用什么技巧来简化公式?

One Small Step
::一小步

Example
::示例示例示例示例

The Apollo 11 was the rocket ship which took the first three men to land on the moon: Neil Armstrong, Edwin “Buzz” Aldrin and Michael Collins. Adria wants to 3D print a model of the Apollo 11 spaceship but does not know if she has enough PLA. She only has one brand new spool of PLA filament which means she can print up to 800 cm ³ . She does not know what the volume of the model rocket ship will be but she is using the shape below to estimate the volume. Based on the dimensions, in centimeters, does Adria have enough PLA filament to print the model rocket ship? Use 3.14 as the value for $\begin{align*}\pi.\end{align*}$
::阿波罗11号是火箭船,搭乘前三名男子登上月球:尼尔·阿姆斯特朗、埃德温·“巴兹”阿尔德林和迈克尔·柯林斯。阿德里亚想用3D打印阿波罗11号飞船的模型,但不知道她是否拥有足够的PLA。她只有一个全新的PLA丝质库,这意味着她可以打印多达800厘米3。她不知道模拟火箭船的体积是多少,但她正在使用下面的形状来估计其体积。根据尺寸,以厘米计,Adria是否有足够的PLA丝质来打印模型火箭船?用3.14作为______的价值。

Use the interactive below to help you visualize the shape of Adria's rocket ship. Do you notice any familiar 3D shapes?
::使用下面的互动来帮助您想象 Adria 火箭飞船的形状。您注意到任何熟悉的 3D 形状吗 ?

As you see in the interactive, a cone is placed on top of a cylinder. This is what is known as a composite solid . A composite solid is one solid made up of two or more solids. To find the volume of a composite solid you must find the volume or the individual parts, a cylinder and cone in this case, and add them together.
::正如您在互动中看到的那样,锥体被放在圆柱体的顶部。这就是所谓的复合固体。复合固体是由两个或两个以上固体组成的一个固体。要找到复合固体的体积,您必须在此情况下找到体积或单个部分、圆锥和锥体,并把它们加在一起。

$\begin{aligned} \begin{array}{ll} Volume of the Cylinder & Volume of the Cone \\ V = π r^{2} h & V = \frac{π r^{2} h}{3} \\ V = 3.14 \cdot 3^{2} \cdot 24 & V = \frac{3.14 \cdot 3^{2} \cdot 12}{3} \\ V = 3.14 \cdot 9 \cdot 24 & V = \frac{3.14 \cdot 9 \cdot 12}{3} \\ V = 3.14 \cdot 216 & V = \frac{3.14 \cdot 108}{3} \\ V = 678.24 & V = \frac{339.12}{3} \\ V = 113.04 \end{array} \end{aligned}$
$\begin{align*}\begin{array} {l | l} \text{Volume of the Cylinder} & \text{Volume of the Cone}\\ \hline V=\pi r^2 h & V = \frac{\pi r^2 h}{3}\\ V=3.14 \cdot 3^2 \cdot 24 & V = \frac{3.14 \cdot 3^2 \cdot 12}{3}\\ V=3.14 \cdot 9 \cdot 24 & V= \frac{3.14 \cdot 9 \cdot 12}{3}\\ V=3.14 \cdot 216 & V = \frac{3.14 \cdot 108}{3}\\ V=678.24 & V = \frac{339.12}{3}\\ & V=113.04 \end{array}\end{align*}$
::ConeVr2hVr2h3V=3.143224V=3.1432123V=3.14924V=3.14924V=3.149123V=3.14216V=3.141083V=6.78.24V=339.123V=113.04。

The total volume of the rocket ship is the volume of the cylinder plus the volume of the cone $\begin{align*}678.24+113.04=791.28 \text{ cm}^3.\end{align*}$
::火箭船的总量是气瓶体积加上锥体体体积678.24+113.04=791.28立方厘米。

Adria h as just enough PLA filament to 3D print the model of the Apollo 11 rocket ship.
::Adria的PLA丝丝足足足足足足足三维打印阿波罗11号火箭船的型号。

Summary
::摘要

The volume of a cone is one-third the volume of a cylinder with the same radius and height, or $\begin{align*}V=\frac{\pi r^2 h}{3}.\end{align*}$
::锥体的体积是圆柱体体体积的三分之一,其半径和高度相同,或Vr2h3。

A composite solid is a solid made up of two or more solids. The volume of a composite solid is the sum of the volumes of each part.
::复合固体是由两个或两个以上固体组成的固体,复合固体的体积是每一部分体积的总和。

Section outline

General

锥体量

Cones ::锥体

Volume of Cones ::锥体量

Using the Formula ::使用公式

One Small Step ::一小步

Summary ::摘要

Cones
::锥体

Volume of Cones
::锥体量

Using the Formula
::使用公式

One Small Step
::一小步

Summary
::摘要