8.1 气瓶量-interactive

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  气瓶量

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  Volume of Objects
  ::物体数量

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   is the amount of space inside a 3-dimensional solid  measured in cubic units . Think of volume as the number of cubic units that can fit inside an object. A cubic unit is a cube of a chosen length. Some examples of cubic units are cubic inches (in ³ ), cubic meters (m ³ ), cubic centimeters (cm ³ ) and cubic feet (ft ³ ). These units are chosen based on the unit system used and the size of the object.
  ::是在立方体单位中测量的三维固体的面积。将体积想象成可以适合一个物体的立方体单位的数量。立方体单位是选定长度的立方体。一些立方体单位的例子有立方英寸(in3)、立方米(m3)、立方厘米(cm3)和立方英尺(ft3),这些单位是根据使用的单位系统和天体大小选择的。

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  Use the interactive below to determine the appropriate cubic units to measure each object.  There may be more than one possible answer. 
  ::使用下面的交互式来确定测量每个对象的适当立方单位。可能有一个以上的答案。

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  A polyhedron is a 3-dimensional solid made of flat sides.  A prism is a polyhedron with 2 parallel and congruent bases. 
  ::多面体是由平面制成的三维固态。棱晶体是一种多面体,有2个平行和相容的基质。

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  To find the volume of a prism, imagine that it can be broken into  unit cubes and counted. A method for doing this is to find the number of cubes along the bottom of the solid  and to multiply that by the number of rows.
  ::要找到棱镜的体积, 想象它可以被拆成单位立方体并进行计算。 这样做的方法就是在固体底部找到立方体的数量, 并乘以行数 。

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  The formula for finding the volume of a  prism is: $\begin{align*}\text{Volume} = \text{Area of the Base} \cdot \text{height}. \end{align*}$  The term “ Area of the Base” is used because the area of the base is equal to the number of cubes that can fit along the base.
  ::查找棱晶体积的公式是:BaseQheight 的卷=区域。使用“基地区域”一词是因为基础区域等于可以与基数相容的立方体数量。

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  Use the interactive below to review this concept. How does changing the number of cubes along the base or height change the volume of the prism?
  ::使用下面的互动来审查这个概念。 如何改变基点或高度上的立方体数量来改变棱晶的体积 ?

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  Volume is an important concept used in many applications . In this lesson, you  will look at how volume  applies to  businesses that use 3D printing .
  ::音量是许多应用中使用的一个重要概念。 在此教训中, 您将查看音量如何适用于使用 3D 打印的企业 。

   

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  Volume of Cylinders
  ::气瓶量

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  The approach you will take to find the volume of a cylinder is very similar to the approach you took to find the volume of a prism. The objective is the same: count the number of cubic units that make up the cylinder. The challenge you face is that the cubes do not fit perfectly into the cylinder. Use the interactive below to explore this concept. 
  ::您要找到圆柱体体积的方法与您找到棱柱体积的方法非常相似。 目标是相同的: 计数圆柱体的立方体单位数。 您面临的挑战是, 立方体不完全适合圆柱体。 使用下面的交互功能来探索这个概念 。

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  Using the Volume Formula for a Cylinder
  ::使用气瓶的音量公式

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  The formula for the volume of a cylinder comes from the formula used to find the volume of a prism. Prisms and cylinders are both solids with parallel and congruent bases. The only difference between a prism and a cylinder is that a prism is a polyhedron, which means it cannot have any curved sides. However, the curved sides of a cylinder do not affect how the volume is determined. Using the formula for the volume of a prism, substitute the area of the base with the area of a circle :
  ::圆柱体体积的公式来自用于查找棱柱体积的公式。棱柱和圆柱体都是有平行和相容基数的固体。棱柱和圆柱体的唯一区别是棱柱和圆柱体的区别是,圆柱体是一个圆柱体,这意味着它不可能有任何弯曲的侧面。然而,圆柱体的曲线侧面不影响如何确定体积。使用棱柱体积的公式,用圆圈区域替代基体区域:

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  $$\begin{align*}&\text{Volume} = \text{Area of Base} \cdot \text{height}\\ &\text{Volume} = \pi \cdot \text{radius}^{2} \cdot \text{height}\end{align*}$$
  ::体积 = baseQheight 区域 Volume radious28

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  Example
  ::示例示例示例示例

  

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  In 2010 and 2007, two massive cylindrical sinkholes opened up in Guatemala City, Guatemala. Both sinkholes were roughly the same size and have been filled with cement. The dimensions of the 2010 sinkhole, in feet, are shown in the interactive below. How much cement would be required to fill the 2010 sinkhole?  
  ::2010年和2007年,在危地马拉的危地马拉市,两个巨大的圆柱形水槽洞被打开。两个水槽洞的大小大致相同,并填满了水泥。下面的交互式显示了2010年水槽洞的足尺寸。要填满2010年水槽洞,需要多少水泥?

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   Steps to Find the Volume of Any Solid
  ::查找任何固体数量的步骤

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  1. Write the formula.
     ::写公式。
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  2. Substitute the dimensions of the solid into the formula.
     ::将固体的维度替换为公式。
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  3. Use order of operations to simplify.
     ::使用操作顺序简化。

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  1. Write the formula.
  ::1. 编写公式。

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  $$\begin{align*}\text{Volume} = \pi \cdot \text{radius}^{2} \cdot \text{height}\end{align*}$$
  ::音量 弧度 2QQQIELL

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  2. Substitute the dimensions of the  solid into the formula.
  ::2. 在公式中取代固体的尺寸。

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  Use 3.14 as the value for  $\begin{align*}\pi.\end{align*}$  The radius is the distance from the center of the circle to the edge , or half the diameter . Therefore, the radius is half of 66, or 33 feet. You are also told that the height of the cylinder is 330 feet.
  ::使用 3. 14 作为 的值。 半径是圆的中心到边缘的距离, 或直径的一半。 因此, 半径为 66 半径, 或 33 英尺。 您也被告知圆柱的高度是 330 英尺 。

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  $$\begin{align*}\text{Volume}=3.14\cdot 33^2 \cdot 330\end{align*}$$  
  ::卷号=3.14332330

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

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  $$\begin{align*}\text{Volume} &= 3.14 \cdot 33^2 \cdot 330\\ \text{Volume} &= 3.14 \cdot 1089 \cdot 330\\ \text{Volume} &= 1,128,421.8\end{align*}$$
  ::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

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  The amount of cement required to fill the hole is 1,128,421.8 ft ³ .
  ::填洞所需的水泥量为1,128,421.8平方英尺。

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  Imagine a cube that is one foot on each side – it would  take 1,128,421.8 cubes to fill the sinkhole.
  ::想象一个每边一英尺的立方体 — — 需要1,128,421.8立方体才能填满坑洞。

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  Example
  ::示例示例示例示例

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  A question may ask you to solve for the volume in terms of $\begin{align*}\pi.\end{align*}$  Treat $\begin{align*}π \end{align*}$  as a variable  that represents 3.14159… Here is how you would solve the previous example in terms of  $\begin{align*}π.\end{align*}$
  ::问题可能要求您用 来解答此音量 。 将 当作代表 3. 14159 的变量处理 。 以下是您如何用 来解答前一例 。

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  $$\begin{align*}\text{Volume} &= π \cdot \text{radius}^{2} \cdot \text{height}\\ \text{Volume} &= π \cdot {33}^{2} \cdot 330\\ \text{Volume} &= π \cdot 1089 \cdot 330\\ \text{Volume} &= π \cdot 359,370\\ \text{Volume} &= 359,370π\end{align*}$$
  ::卷号332330 卷号1089330 卷号359370 卷号359370

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  The a mount of cement required to fill the hole   is   $\begin{align*}359,370π\end{align*}$  ft ³ .
  ::填洞所需的水泥量为359,370平方英尺。

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  Why do you think you would solve for a volume in terms of  $\begin{align*}\pi \end{align*}$  rather than using 3.14 or some other rounded  value? 
  ::为什么你认为你会用... 而不是3.14 或其他四舍五入的数值 来解答一个音量呢?

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  Volume and 3D Printing
  ::卷卷和3D印刷

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  3D Printing is reshaping the way that things are made in the modern world. By designing a prototype of an object on a computer, a 3D printer can bring it into the real world. However, the process and decisions involved are not that simple. Every object needs to be made with the printing material and that material costs money. The more volume an object has, the greater the material cost. 
  ::3D 打印正在改变现代世界中制造事物的方式。 通过设计计算机上物体的原型, 3D 打印机可以将它带入现实世界。 但是, 所涉及的过程和决定并不那么简单。 每个物体都需要用印刷材料制作, 材料成本也很高。 一个物体的体积越大, 材料成本就越高 。

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  Example
  ::示例示例示例示例

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  One of the main materials used in 3D printing is polylactide (PLA) filament. The average cost of PLA filament is about $19 for 800 cubic centimeters.  If you need to print the solid cylinder shown below, what will the material cost be? Use 3.14 as the value for  $\begin{align*}\pi. \end{align*}$
  ::3D打印中使用的主要材料之一是多晶化(PLA)丝质。PLA丝质的平均成本为800立方厘米约19美元。如果需要打印以下显示的固体圆柱体,那么材料成本是多少?使用3.14的值来计算 。

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  To begin, you must find the volume of the cylinder , which is  the amount of material that will be needed to print the cylinder.  You can figure this out  by following the steps outlined in the previous activity.
  ::首先,您必须找到圆柱体的体积,也就是打印圆柱体所需的材料量。您可以按照上一个活动概述的步骤找到这一点。

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  1. Write the formula.
  ::1. 编写公式。

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  $$\begin{align*}\text{Volume} = \pi \cdot \text{radius}^{2} \cdot \text{height}\end{align*}$$
  ::音量 弧度 2QQQIELL

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  2. Substitute the dimensions of the  solid into the formula. Use 3.14 as the value for  $\begin{align*}\pi.\end{align*}$
  ::2. 在公式中以固体尺寸代替固体尺寸,使用3.14作为______的值。

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  $$\begin{align*}\text{Volume} = 3.14 \cdot 0.5^2 \cdot 12\end{align*}$$
  ::体积=3.140.5212

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

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  $$\begin{align*}\text{Volume} &= 3.14 \cdot 0.5^2 \cdot 12\\ \text{Volume} &= 3.14 \cdot 0.25 \cdot 12\\ \text{Volume} &= 9.42\end{align*}$$
  ::卷号=3.140.5212 卷号=3.140.2512 卷号=9.42

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  The volume of the cylinder is 9.42 cm ³ .
  ::气瓶的体积为9.42厘米3。

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  One way to determine the material cost  is to use  a proportion with the price to volume ratios:
  ::确定物质成本的一个方法,是使用价格与数量比率的比例:

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  $$\begin{align*}\frac{$19.19}{800 \text{ cm}^{3}} &= \frac{x}{9.42 \text{ cm}^{3}}\\\\ 19.19\cdot 9.42 &= 800x\\ 178.98 &= 800x\\ 0.223725 &= x\end{align*}$$
  ::191,800 cm3=x9.42 cm319.19.9.42=800x178.98=800x0.223725=x

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  Rounded to the nearest cent, it will cost $0.22 in  PLA filament to print the cylinder.
  ::在四舍五入到最近的百分比后,打印圆筒的PLA丝质将花费0.22美元。

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   Summary
  ::摘要

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  • Volume is the amount of space inside a 3D solid measured in cubic units.
    ::体积是指以立方体单位测量的3D固体内的空间量。
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  • A prism is a solid made of flat sides with two parallel and congruent bases.  Its volume  is found by multiplying the area of  the base  by   the height, or  $\begin{align*}V=B\cdot h.\end{align*}$  
    ::棱晶是由平面制成的固态, 具有两个平行和一致的基点。 它的体积通过将基点面积乘以高度或 V=Bh 来发现 。
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  • A cylinder is a solid with two parallel and congruent circular bases. Its volume  is found by multiplying the area of the circle by  the height, or  $\begin{align*}V=\pi r^2h.\end{align*}$  
    ::一个圆柱体是一个固体,有两个平行和一致的圆圆基。其体积通过圆圈区域乘以高度或Vr2h而发现。