寻找锥体的方方面面

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Flow Rates
::流动率

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Finding the of a cone is useful for figuring out how much space an object takes up. This information is important  for building a structure or calculating how much of something, say, ice cream that a cone can hold.  However, when the volume needs to be a  specified amount, you need to find the dimensions that  result in that amount.  You will learn how to find the dimensions of a cone given the volume.
::找到锥体有助于了解一个物体占用多少空间。 此信息对于构建一个结构或计算锥体能持有多少冰淇淋非常重要。 但是, 当数量需要指定数量时, 您需要找到导致该数量的尺寸。 您将学习如何根据数量找到锥体的尺寸 。

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One common application of cones in the real world is flow rates through conical funnels. T o find the rate at which a liquid flows through a funnel, you need to use the volume formula. While the applications of flow rate are mostly used in calculus, you can use your knowledge of similar triangles and proportions to understand the basics. This concept will be explored more later in the lesson.
::圆锥体在现实世界中的一种常见应用是通过圆锥漏斗的流速。要找到液流通过漏斗的速率,您需要使用体积公式。虽然流速的应用大多用于微积分,但您可以使用对类似三角形和比例的了解来理解基本情况。这个概念将在课后更深入地探讨。

 

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Solving for the Height
::高度的解决方案

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The approach to finding missing dimensions of a cone will be similar to how you found missing dimensions of a cylinder . The only difference is that the volume of a cone formula involved dividing by 3. To address this, the first step when solving is to multiply both sides of the equation by 3.
::找到锥体缺失维度的方法与您如何发现圆柱体缺失维度的方法相似。 唯一的区别是锥体公式的体积除以3。 要解决这个问题,解决问题的第一步是将方程式的两侧乘以3。

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Example
::示例示例示例示例

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Older water tower designs have a cylindrical design with a cone-shaped roof on top. A water tower has a cone on top with the dimensions, in feet, shown in the interactive below. Determine the height of the roof of the water tower. 
::旧水塔的设计有一个圆柱形设计,顶部有一个锥形的屋顶。水塔顶上有一个锥形的锥体,其尺寸在脚上,在下面交互式显示。确定水塔顶部的高度。

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1. Write the formula.
::1. 编写公式。

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

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2. Substitute the dimensions of the  cone into the formula. Use 3.14 as the value for $\begin{align*}\pi\end{align*}$ .
::2. 将锥锥体的尺寸替换为公式,使用3.14作为 的值。

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$$\begin{align*}1,884 = \frac{3.14 \cdot {10}^{2} \cdot h}{3}\end{align*}$$
::1,884=3.1410°h3

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3. Solve the equation for the variable.
::3. 解决变量的方程。

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$$\begin{align*}1,884 &= \frac{3.14 \cdot {10}^{2} \cdot h}{3} \\ 5,652 &= 3.14 \cdot {10}^{2} \cdot h \\ 5,652 &= 3.14 \cdot 100 \cdot h \\ 5,652 &= 314h \\ 18 &= h\end{align*}$$
::1,884=3.14102h35652=3.14102h5652=3.1410h5652=3.14100h5652=314h18=h

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The height of the cone is 18 feet.
::锥形的高度是18英尺

 

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Solving for the Radius
::半径的解决

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Example
::示例示例示例示例

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A conical pine tree has the approximate dimensions, in feet, displayed in the interactive below. Determine the radius of the tree from the given dimensions.
::锥形松树的足部大致尺寸显示在下面交互的下方。从给定尺寸中确定树的半径。

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1. Write the formula.
::1. 编写公式。

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

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2. Substitute the dimensions of the  cone into the formula. Use 3.14 as the value for $\begin{align*}\pi\end{align*}$  .
::2. 将锥锥体的尺寸替换为公式,使用3.14作为 的值。

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$$\begin{align*}785 = \frac{3.14 \cdot {r}^{2} \cdot 30}{3}\end{align*}$$
::785=3.14r2303

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3. Solve the equation for the variable.
::3. 解决变量的方程。

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$$\begin{align*}785 &= \frac{3.14 \cdot {r}^{2} \cdot 30}{3} \\ 2,355 &= 3.14 \cdot {r}^{2} \cdot 30 \\ 2,355 &= 3.14 \cdot {r}^{2} \cdot 30 \\ 2,355 &= 94.2{r}^{2} \\ 25 &= {r}^{2}\end{align*}$$
::785=3.14r2}3032,355=3.14}r2}302,355=3.14}r2}302,355=3.14}r2}302,355=94.2r2=r2

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The equation  $\begin{align*}{r}^{2} = 25\end{align*}$  states that some number multiplied by itself is equal to 25. If you multiply whole numbers by themselves in your head, you should realize that $\begin{align*}5 \cdot 5 = 25\end{align*}$ .
::方程 r2=25 表示,某些数字本身乘以25等于25,如果在你的脑中自己将整个数字乘以,你应该意识到5=5=25。

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The radius of the cone is 5 feet.
::锥形的半径是5英尺

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Flow Rates Continued
::流动率

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A conical funnel has the dimensions displayed below.  When water is  in the funnel, the water forms a cone. During the draining process, the volume of the water decreases but the shape remains a cone.
::锥形漏流具有以下尺寸。 当水在漏流中时, 水会形成锥体。 在排水过程中, 水的体积会下降, 但形状仍然是锥体 。

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Example
::示例示例示例示例

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The water in a funnel forms a cone with volume 452.16 cm ³ . What are the dimensions of the cone?
::漏斗中的水形成一个圆锥体,卷积452.16厘米。 锥体的尺寸是多少?

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As shown in the interactive, the height and radius of the water form similar triangles based on the . The height will always be double the radius at every point in the draining process. This relationship can be expressed using the equation :  $\begin{align*}h = 2r\end{align*}$
::如交互式中显示的,水的高度和半径形成基于 . 的相似三角形。 高度总是排水过程中每个点半径的两倍。 此关系可以用公式 h=2r 来表达 。

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1. S ubstitute  $\begin{align*}2r \end{align*}$  into the volume of a cone equation for  height   so that you can solve for radius .

$$\begin{align*}\text{Volume} & = \frac{\pi \cdot r^2 \cdot h}{3}\\ &= \frac{\pi \cdot {r}^{2} \cdot (2r)}{3}\end{align*}$$
::1. 将 2r 替换为角形方程式的体积, 以便您能够解析半径 。Volumer2h3r2(2r)3

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2. S ubstitute the known dimensions. Use 3.14 as the value for $\begin{align*}\pi\end{align*}$ .
::2. 替代已知尺寸,使用3.14作为 的值。

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$$\begin{align*}452.16 &= \frac{3.14 \cdot {r}^{2} \cdot (2r)}{3}\end{align*}$$
::452.16=3.14r2(2r)3

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3. R earrange this expression using the commutative property and write  $\begin{align*}r\cdot r\cdot r\end{align*}$   in exponential form as  $\begin{align*} {r}^{3}.\end{align*}$   Then solve for  $\begin{align*}r:\end{align*}$  
::3. 使用通量属性重新排列该表达式,并以指数形式将 rrrr 写成 r3。 然后解决 r:

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$$\begin{align*}452.16 &= \frac{3.14 \cdot r \cdot r \cdot 2 \cdot r}{3}\\ 452.16 &= \frac{3.14 \cdot 2 \cdot r \cdot r \cdot r}{3} \\ 452.16 &= \frac{6.28 {r}^{3}}{3} \\ 1,356.48 &= 6.28 {r}^{3} \\ 216 &= {r}^{3}\end{align*}$$
::452.16=3.14r}2}r3452.16=3.142r}r}3452.16=6.28r}31356.48=6.28r 3216=r3

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You have not yet learned the opposite operation of cubing a number. However, you can figure it out by understanding what the third power means. The equation  $\begin{align*}{r}^{3} = 216\end{align*}$  means that some number multiplied by itself three times equals 216. If you go through the whole numbers, it will not take long to find that $\begin{align*}6 \cdot 6 \cdot 6 = 216\end{align*}$ .
::您还没有学到对一个数字进行截取的相反操作。 但是, 您可以通过理解第三个功率意味着什么来理解它。 等式 r3=216 意味着某些数字本身乘以3倍等于 216 。 如果您浏览整个数字, 很快就会发现 66=216 。

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T he radius of the water is 6 cm. The diameter is double the radius which means that the diameter will be $\begin{align*}6 \cdot 2 = 12\text{ cm.}\end{align*}$
::水的半径为6厘米。直径是半径的两倍,这意味着直径将为6×2=12厘米。