8.6 球体体体积-interactive

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  Spheres
  ::球球面

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  Spheres are an extremely important solid. A  sphere is one of the most common shapes in the universe. Planets, suns, moons, and anything with a strong enough gravitational force will take on the shape of a sphere. The force of gravity which comes from the center of mass pulls equally in all directions. Imagine squeezing a snowball in your hands;  if the pressure is equal in all directions, it will result in a sphere.
  ::球体是一个极为重要的固体体。球体是宇宙中最常见的形状之一。行星、太阳、月亮和任何具有足够强大引力的物体,都会以球体的形式出现。来自质量中心的引力向各个方向平均拉动。想象一下在你的手中挤压雪球;如果压力在所有方向上相等,它将导致球体。

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  Spheres are also important because they have the most per (see   for a review of surface area). The reverse is also true,  in that they have the least surface area per volume. If you took three different solids with equal surface areas, the sphere would have more volume than the other two solids .
  ::球体之所以重要,也是因为它们的每个球体最多(请参见对表面积的审查)。反之亦然,因为它们每个体积的表面积最小。如果取出三个不同的、表面积相等的固体,则球体的体积会比其他两个固体的体积要大。

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  In science, reaction times are influenced by the surface area of a solid –  t he greater the surface area, the faster the reaction. A simple example involves melting ice. Between  an ice cube and an ice sphere with equal volume, which would melt first ?
  ::在科学中,反应时间受固体表面面积的影响 — — 表面面积越大,反应越快。 一个简单的例子就是冰融化。 在冰块和体积相等的冰球之间,首先融化的是一个冰块和体积相等的冰球?

   

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  Volume of Spheres
  ::环球音量

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  As with the other shapes, you are going to have to be clever to determine how many cubic units can fit in a sphere. You can use the cube’s relation to a circumscribed cylinder to determine its volume. Circumscribed means constructed around a shape touching as many points as possible. Use the interactive below to explore this relationship.
  ::与其他形状一样, 您需要聪明地确定一个球体中有多少立方体单位。 您可以使用立方体与受限制的圆柱体的关系来决定其体积。 环形环绕形状构造, 触摸到尽可能多的点。 使用下面的互动来探索这种关系 。

  

  INTERACTIVE

  Sphere vs Cylinder and Cone

  

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  • Drag the slider to change the radius (and heights) of the sphere, cylinder, and cone.
    ::拖动滑块以改变球体、圆柱体和锥体的半径(和高度) 。
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  • Drag the red points to translate the shapes and compare their volumes.
    ::拖曳红色点以翻译形状并比较其体积。

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  Using the Formula
  ::使用公式

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  As you saw in the interactive above, the  volume of a sphere will always be two-thirds the volume of a circumscribed cylinder. You can u se t he formula for the volume of a cylinder,  $\begin{array}{r}\pi \cdot {\text{radius}}^2\cdot \text{height},\end{array}$   to find the formula for the volume of a sphere:
  ::正如您在上文互动中看到的那样,球体的体积将始终是受限制圆柱体体体积的三分之二。您可以使用圆柱体体体积的公式 radus2QQhleight, 找到球体体体体积的公式 :

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  $$\begin{array}{r}\begin{array}{ll}\text{Volume of a Sphere}& \text{Explanation}\\ V=\frac{2}{3}(\pi \cdot {\text{radius}}^2\cdot \text{height})& \text{The volume of a sphere is two-thirds the volume of a cylinder.}\\ V=\frac{2}{3}(\pi \cdot {\text{radius}}^2\cdot 2\cdot \text{radius})& \text{The height is equal to the diameter or twice the radius.}\\ V=\frac{2\cdot 2\cdot \pi \cdot \text{radius}\cdot \text{radius}\cdot \text{radius}}{3}& \text{Rearrange using the commutative property.}\\ V=\frac{4\cdot \pi \cdot {\text{radius}}^3}{3}& \text{Simplify.}\end{array}\end{array}$$

  
  ::圆柱体的体积是圆柱体体体积的三分之二。V=23(radius222radius) 高度等于直径或半径的两倍。V=22radiusradius3Rearrange,使用移动特性。V=4radius33Simpligiation。

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  Example
  ::示例示例示例示例

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  A snow globe has the dimensions , in inches, displayed below. Find the volume of the snow globe. Use 3.14 as the value for $\begin{array}{r}\pi \end{array}$ .
  ::雪球的尺寸为英寸, 显示在下面。 查找雪球的体积。 使用 3. 14 作为 ° 的值 。

  

  INTERACTIVE

  Snow Globe

  

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  • Click the radius to reveal its length.
    ::单击半径以显示其长度 。
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  • Drag the screen to change your point of view.  Use two fingers on your trackpad or touch screen or scroll the mouse wheel to zoom.
    ::拖曳屏幕以改变您的视图点。 在您的跟踪板上使用两根手指或触摸屏幕或滚动鼠标方向盘以缩放 。

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  1. Write the formula.
  ::1. 编写公式。

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  $$\begin{array}{r}\text{Volume}=\frac{4\cdot \pi \cdot {\text{radius}}^3}{3}\end{array}$$

  
  ::体积=4半径33

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  2. Substitute the dimensions of the  snow globe into the formula. Use 3.14 as the value for $\begin{array}{r}\pi \end{array}$ .
  ::2. 将雪球的尺寸替换为公式,使用3.14作为______的值。

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  $$\begin{array}{r}\text{Volume}=\frac{4\cdot 3.14\cdot 6^3}{3}\end{array}$$

  
  ::卷号=43.146333

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

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  $$\begin{array}{rl}\text{Volume}& =\frac{4\cdot 3.14\cdot 6^3}{3}\\ \text{Volume}& =\frac{4\cdot 3.14\cdot 216}{3}\\ \text{Volume}& =\frac{864\cdot 3.14}{3}\\ \text{Volume}& =\frac{2,712.96}{3}\\ \text{Volume}& =904.32\end{array}$$

  
  ::卷=43.142163 卷=8643.143 卷=2,712.963 卷=904.32

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  The volume of the snow globe is 904.32 cm ³ .
  ::雪球的体积是904.32厘米3。

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  Discussion Question
  ::讨论问题

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  Are there any tricks we could have used to simplify the formula?
  ::是否有任何我们本可以用来简化公式的花招?

     

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  Life on Mars
  ::火星上的生命

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  Spheres are also important in measuring domes and hemispheres which are half spheres. These designs are very common in architecture.
  ::球体在测量作为半球体的穹顶和半球方面也很重要,这些设计在建筑中非常常见。

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  Example
  ::示例示例示例示例

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  Recently, six scientists lived in a dome for an entire year in Hawaii to simulate the environment on a mission to Mars. Use the dimensions of the dome, in feet, shown below to find the volume of the oxygen that the dome could contain. Use 3.14 as the value for $\begin{array}{r}\pi \end{array}$ .
  ::最近,有6名科学家在夏威夷的一个圆顶中生活了整整一年,以模拟火星飞行任务的环境。 使用下面显示的圆顶的足尺寸来寻找圆顶可能含有的氧气量。 使用3.14作为______的值。

  

  INTERACTIVE

  Life on Mars

  

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  • Click the diameter to reveal its length.
    ::点击直径以显示其长度 。
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  • Drag the screen to change your point of view.  Use two fingers on your trackpad or touch screen or scroll the mouse wheel to zoom.
    ::拖曳屏幕以改变您的视图点。 在您的跟踪板上使用两根手指或触摸屏幕或滚动鼠标方向盘以缩放 。

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  Since oxygen expands to fit its container, the volume of the oxygen is equal to the volume of the dome. Use the formula  for  volume of a sphere to find the  volume of the dome.
  ::由于氧气膨胀以适合其容器,氧气的体积等于圆顶的体积。使用球体体体积的公式来找到圆顶的体积。

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  1. Write the formula.
  ::1. 编写公式。

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  $$\begin{array}{r}\text{Volume}=\frac{4\cdot \pi \cdot {\text{radius}}^3}{3}\end{array}$$

  
  ::体积=4半径33

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  2. Substitute the dimensions of the  dome  into the formula. Use 3.14 as the value for $\begin{array}{r}\pi \end{array}$  and 18 feet as the radius because it is half the diameter of 36 feet.
  ::2. 将圆顶的尺寸替换为公式,使用3.14的值,以...和18英尺的值作为半径,因为它的直径为36英尺的一半。

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  $$\begin{array}{r}\text{Volume}=\frac{4\cdot 3.14\cdot {18}^3}{3}\end{array}$$

  
  ::卷号=43.141833

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

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  $$\begin{array}{rl}\text{Volume}& =\frac{4\cdot 3.14\cdot {18}^3}{3}\\ \text{Volume}& =\frac{4\cdot 3.14\cdot 5,832}{3}\\ \text{Volume}& =\frac{23,328\cdot 3.14}{3}\\ \text{Volume}& =\frac{73,249.92}{3}\\ \text{Volume}& =24,416.64\end{array}$$

  
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  The volume of the dome is  half the volume of the sphere, so the volume of oxygen that  the dome could contain is  12,208.32  ft ³ .
  ::圆顶的体积是球体体体积的一半, 所以圆顶可能含有的氧气体积是12,208.32英尺。

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  Summary
  ::摘要

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  • The volume of a sphere is two-thirds the volume of a circumscribed cylinder, or  $\begin{array}{r}V=\frac{4}{3}\pi r^3.\end{array}$
    ::球体的体积是受限制气瓶的体积的三分之二,或V=43°r3。
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  • The volume of a hemisphere is half the volume of a sphere, or  $\begin{array}{r}V=\frac{2}{3}\pi r^3.\end{array}$
    ::一个半球的体积是一个球体体体积的一半,或V=23°r3。