Section: 查找球体的方方面面 | 8.7 寻找球体的方方面面-interactive

Section outline

Missing Dimensions of Spheres
::球体的缺失尺寸

In this lesson , you will be learning how to find missing dimensions of a sphere . Even though a sphere has three dimensions, you only need one variable to know every property of the sphere. There are only two variables in the formula – if you know the radius , then you can find the volume and vice versa.
::在此教训中, 您将会学习如何找到一个球体的缺失维度。即使一个球体有三个维度, 您只需要一个变量才能知道球体的每一个属性。公式中只有两个变量 — — 如果您知道方圆, 那么您就可以找到体积, 反之亦然。

Discussion Questions
::讨论问题讨论问题

How can you use the radius to find the diameter of the sphere?
::您如何使用半径找到球体的直径 ?

How can you use the radius to find the length, width, and height of the sphere?
::您如何使用半径查找球体的长度、宽度和高度 ?

How can the radius be used to find the of the sphere?
::如何利用半径找到球体?

How can you use the volume to find the radius?
::你如何用体积找到半径?

Solving for the Radius
::半径的解决

As stated above, the volume is all you need to find the radius of a sphere.
::如上所述,体积是找到一个球体半径所需要的全部。

Example
::示例示例示例示例

The volume of a tennis ball is roughly 113.04 cm ³ . W hat is the radius of the tennis ball? Use 3.14 as the value for $\begin{aligned} π . \end{aligned}$
::网球的体积大约是113.04厘米。网球的半径是多少?使用3.14作为的值。

1. Write the formula.
::1. 编写公式。

$\begin{aligned} Volume = \frac{4 \cdot π \cdot {radius}^{3}}{3} \end{aligned}$

::体积=4半径33

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

$\begin{aligned} 113.04 = \frac{4 \cdot 3.14 \cdot r^{3}}{3} \end{aligned}$

::113.04=43.1433

3. Solve the equation for the missing variable.
::3. 解决缺失变量的方程。

$\begin{aligned} 113.04 & = \frac{4 \cdot 3.14 \cdot r^{3}}{3} \\ 339.12 & = 4 \cdot 3.14 \cdot r^{3} \\ 339.12 & = 12.56 \cdot r^{3} \\ 27 & = r^{3} \end{aligned}$

::113.04=43.14r333.39.12=43.14r33.39.12=12.56r327=r3

The equation $\begin{aligned} r^{3} = 27 \end{aligned}$ means that some number multiplied by itself three times is equal to 27. If you go through the list of whole numbers, multiplying each one by itself three times, you will eventually get to $\begin{aligned} 3 \cdot 3 \cdot 3 = 27 \end{aligned}$ . This means that $\begin{aligned} r \end{aligned}$ is equal to 3.
::方程 r3=27 表示一些数乘以3 本身等于 27 。如果通过整个数列表,每个数本身乘以3 3 =3 = 27, 则表示 r 等于 3。这意味着 r 等于 3 = 3 = 3 = 3 = 3 = 27 。

The radius of a tennis ball is 3 cm.
::网球的半径是3厘米

Aquarium Design
::水族设计

Example
::示例示例示例示例

Jellyfish Tank

Maggie is designing a spherical jellyfish tank for a local aquarium similar to the one pictured above. When speaking with a marine biologist, she learned that the tank will need to hold 14,130 cubic feet of water so that the jellyfish will have enough room to move. The tank itself will sit on a display. To build the display, Maggie needs to know the length, width, and height of the tank. Since the tank is a sphere, the length, width, and height will all be equal, so she only needs to find the height. What is the height of the tank?
::Maggie正在设计一个球形水母罐,用于一个与上面所描述的类似的地方水族馆。当她与海洋生物学家交谈时,她了解到水罐需要持有14,130立方英尺的水,以便水母有足够的活动空间。水罐本身会坐在一个显示器上。要建立显示器,Maggie需要了解罐体的长度、宽度和高度。由于水箱是一个球体,其长度、宽度和高度都是一样的,所以她只需要找到高度。水罐体的高度是多少?

To find the height of the tank, Maggie will need to find the radius first. Use the same steps that you used in the above example.
::要找到坦克的高度, Maggie 需要先找到半径。使用与您在以上示例中使用的相同的步骤。

$\begin{aligned} Volume & = \frac{4 \cdot π \cdot {radius}^{3}}{3} \\ 14, 130 & = \frac{4 \cdot 3.14 \cdot r^{3}}{3} \\ 42, 390 & = 4 \cdot 3.14 \cdot r^{3} \\ 42, 390 & = 12.56 \cdot r^{3} \\ 3, 375 & = r^{3} \end{aligned}$

::体积=4radious3314,130=43.14r3342,390=43.14r342,390=12.56r33375=r3

The equation $\begin{aligned} r^{3} = 3, 375 \end{aligned}$ states that some number multiplied by itself three times is equal to 3,375. Guessing and checking will be difficult to do by hand, but eventually, you would get to $\begin{aligned} 15 \cdot 15 \cdot 15 = 3, 375 \end{aligned}$ . This means that $\begin{aligned} r \end{aligned}$ is equal to 15. T he height of the tank is equal to the diameter, so you need to double the radius.
::方程 r3 = 3 375 表示某些数字本身乘以3 3 375 等于 3 375 。猜测和检查将难以手动进行, 但最终会达到 15 15 15 = 3 375 。这意味着 r 等于 15 。这意味着 r 等于 15 。罐体的高度等于直径, 因此您需要将半径增加一倍。

The height of the tank is $\begin{aligned} 2 (15) = 30 \end{aligned}$ ft.
::罐体的高度为2(15)=30英尺。

How Radius Affects Volume
::半半径如何影响音量

The lesson discussed how the dimensions affected the volume of a pyramid . A sphere only has one variable in the formula, the radius. Use the interactive below to see the radius affects the volume of a sphere .
::课程讨论了维度如何影响金字塔的体积。球体在公式中只有一个变量, 即半径。使用下面的交互来查看半径如何影响球体的体积。

INTERACTIVE

Volume and Radius

Move the slider to watch the sphere change size.
::移动滑动器以观看球体变化大小。

Press the buttons to view different perspectives.
::按下按钮查看不同角度。

Each button will have two perspectives, one the reflection of the other.
::每个按钮将有两个视角, 一个是另一个的反射。

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Discussion Questions
::讨论问题讨论问题

Set the slider to 2. How did doubling the radius affect the volume of the sphere? Why did this happen?
::设置滑块到 2 。半径加倍如何影响球体的体积 ? 为什么发生这种情况 ?

How do you think tripling the radius will affect the volume of the sphere? Why?
::你认为方圆三倍会影响球体的体积?

How do you think cutting the radius in half will affect the volume of the sphere? Why?
::你认为把半径切成两半会影响球体的体积吗?

Summary
::摘要

To find the radius of a sphere , substitute volume into the volume formula and solve for the radius.
::要找到一个球的半径, 请在音量公式中替换音量, 并解决半径。

When the radius of a sphere is scaled by some factor $\begin{aligned} k, \end{aligned}$ the volume will be scaled by $\begin{aligned} k^{3} . \end{aligned}$
::当一个球体的半径以某种系数 k 缩放时, 音量将用 k3 缩放。

Section outline

Missing Dimensions of Spheres ::球体的缺失尺寸

Solving for the Radius ::半径的解决

Aquarium Design ::水族设计

How Radius Affects Volume ::半半径如何影响音量

Summary ::摘要

Missing Dimensions of Spheres
::球体的缺失尺寸

Solving for the Radius
::半径的解决

Aquarium Design
::水族设计

How Radius Affects Volume
::半半径如何影响音量

Summary
::摘要