寻找圆柱体的失踪尺寸

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Energy Drink Design
::能源饮料设计

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Picture a soda can. Soda cans are generally all the same shape and have the same general dimensions . Over the past two to three decades, energy drinks have risen in popularity. When you think of an energy drink can, it is likely easy to picture. However, despite containing the same amount of liquid as a soda can and having the same cylindrical shape, the can you imaged probably had distinctly different dimensions. The dimensions of a soda can have a surprisingly significant impact on both sales and profit. Why do you think energy drinks tend to be longer and narrower?
::苏打汽水可以。 苏打罐一般形状相同, 并且具有相同的通用尺寸。 在过去的二到三十年中, 能量饮料越来越受欢迎。 当你想到能量饮料可以, 它可能很容易被描绘出来。 然而, 尽管含有与苏打罐一样数量的液体, 并且具有相同的圆柱形形状, 你所想象的液体可能具有截然不同的尺寸。 苏打罐的尺寸可能会对销售和利润产生惊人的重大影响。 为什么你认为能量饮料会长而窄?

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Solving for the Height
::高度的解决方案

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The dimensions of a cylinder are determined by the restrictions on the , height , and diameter . There are many situations where a solid must have a specific volume but the dimensions to get that volume can be manipulated within a restricted amount. These are the situations that you will be exploring over the next few examples.
::圆柱体的尺寸是由对圆柱体、高度和直径的限制决定的。在很多情况下,固体必须有一个特定的体积,但获得这种体积的尺寸可以在一定数量内操纵。下面几个例子中,你将会探讨这些情况。

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Example
::示例示例示例示例

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A cylindrical grain silo has the dimensions provided below. Use the dimensions to find the height of the silo.
::圆柱形谷物筒仓具有以下尺寸。 使用该尺寸找到筒仓的高度 。

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To find either the height or radius , you will use steps very similar to the ones used in the previous chapter.
::要找到高度或半径, 您将使用与上一章所用的步骤非常相似的步骤 。

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1. Write the formula.
::1. 编写公式。

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

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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。

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$$\begin{align*}3,014.4 = 3.14 \cdot {8}^{2} \cdot h\end{align*}$$
::3,014.4=3.1482h

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

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$$\begin{align*}3,014.4 &= 3.14 \cdot {8}^{2} \cdot h \\ 3,014.4 &= 3.14 \cdot 64 \cdot h \\ 3,014.4 &= 200.96 \cdot h\\ 15 &= h\end{align*}$$
::3,014.4=3.1482h3,014.4=3.1464h3,014.4=200.96h15=h

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The height of the cylindrical grain silo is 15 ft.

::圆柱形谷物筒仓的高度是15英尺。

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You could have solved for the height in terms of $\begin{align*}\pi\end{align*}$  like so:
::你可以用这样的方式解决高度问题,比如:

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$$\begin{align*}\text{Volume} = π \cdot {r}^{2} \cdot h \\ 3,014.4 = π \cdot {8}^{2} \cdot h \\ 3,014.4 = π \cdot 64 \cdot h \\ 3,014.4 = 64π\cdot h \\ \frac{3,014.4}{64π} = h \\ \frac{47.1}{π} = h \end{align*}$$
::体积2h3,014.482h3,014.464h3,014.4=64h3,014.464h47.1h

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The height of the cylindrical grain silo in terms of $\begin{align*}\pi\end{align*}$  is $\begin{align*}\frac{47.1}{π}\:\text{ ft}^{3}.\end{align*}$
::圆柱体粒子井的高度,按 = = 47.1 平方英尺计算,为 47.1 平方英尺。

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Solving for the Radius
::半径的解决

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Solving for the radius becomes a little more complicated because you have never solved for a number which has been squared. Addition and subtraction are opposites. Multiplication and division are opposites. But what is the opposite of squaring a number? Look at a simple example:
::半径的解析变得略为复杂一些, 因为您从未解答一个平方数。 增减是相反的。 乘法和除法是相反的。 但是, 乘法和除法是相反的。 但是, 乘法和除法是相反的吗? 看看一个简单的例子 :

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$$\begin{align*}{x}^{2} = 16\end{align*}$$
::x2=16

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One approach to figuring this out is to multiply the whole numbers by themselves until you get 16:
::一种办法就是自己乘以整个数字 直到你得到16:

$$\begin{align*}1\cdot 1&= 1\\ 2\cdot2 &= 4\\ 3\cdot3 &= 9\\ 4\cdot4 &= 16\end{align*}$$

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The solution to the equation is  $\begin{align*}x = 4.\end{align*}$
::方程式的解决方案是 x=4 。

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Example
::示例示例示例示例

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A cylindrical battery has the dimensions provided below. Use the dimensions to find the diameter of the battery. Use 3.14 as the value for $\begin{align*}\pi \end{align*}$ . 
::圆柱形电池组具有以下尺寸。使用这些尺寸来找到电池组的直径。使用3.14的数值作为 。

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This question asks for the diameter instead of the radius. Since the volume of a cylinder formula involves the radius, you can solve for the radius and double it to find the diameter.
::这个问题要求直径而不是半径。 由于圆柱形公式的体积涉及半径, 您可以解析半径, 并将其翻倍以找到直径 。

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1. Write the formula.
::1. 编写公式。

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

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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。

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$$\begin{align*}791.28 = 3.14 \cdot {r}^{2} \cdot 28\end{align*}$$
::791.28=3.14r228

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

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$$\begin{align*}791.28 &= 3.14 \cdot {r}^{2} \cdot 28 \\ 791.28 &= 3.14 \cdot {r}^{2} \cdot 28 \\ 791.28 &= 87.92{r}^{2} \\ 9 &= {r}^{2}\end{align*}$$
::791.28=3.14r228791.28=3.14r228791.28=87.92r29=r2

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The equation  $\begin{align*}{r}^{2} = 9\end{align*}$  asks you to find the number that when squared is equal to 9. If you multiply the whole numbers by themselves, you will quickly see that $\begin{align*}3\cdot 3 = 9\end{align*}$ . This means that the radius is 3 mm. To find the diameter, double the radius.
::方程式 r2=9 要求您找到当平方等于 9 的数值。 如果您自己将整数乘以 9, 您将很快看到 33=9 。 这意味着半径为 3 毫米。 要找到直径, 请将半径翻一番 。

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The diameter of the cylindrical battery is 6 mm.
::圆柱形电池的直径为6毫米。

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Discussion Question
::讨论问题

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How would you figure out a radius if you are not given a perfect square ?
::如果你不给一个完美的方形 你会怎么弄出半径呢?

   

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Energy Drink Design Continued
::继续能源饮料设计

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Use the interactive below to see how changes in the radius and height of a cylinder can affect the volume. You can even use the interactive to help you in the next design challenge!
::使用下面的交互效果来查看圆柱体半径和高度的变化如何影响音量。您甚至可以使用交互效果来帮助您迎接下一个设计挑战 !

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You design it!
::你设计它!

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Imagine that you have been tasked with creating a new energy drink. You are responsible for naming the drink and designing the can. The can needs to be able to hold 12 ounces of liquid , which is  equivalent to approximately 354.88 cubic centimeters. To add extra space for air in the can, the volume needs to be 360 cubic centimeters. You are responsible for designing the dimensions of the can. Choose the radius first and then solve for the accompanying height. Use 3.14 as a value for  $\begin{align*}\pi.\end{align*}$  
::想象一下您被赋予创建新的能量饮料的任务。 您负责命名饮料名称和设计罐。 罐头需要能够持有12盎司的液体, 相当于约354. 88立方厘米。 要在罐头中添加额外的空气空间, 容量需要360立方厘米。 您负责设计罐头的尺寸。 首先选择半径, 然后解决相应的高度。 使用 3. 14 作为 的值 。