课程： 10.13 气瓶量定量高地

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气瓶体积高地
Gregory's family just bought a hot tub for their lake house. The hot tub company told his family that the tub holds 125 cubic feet of water. Gregory is interested to know how deep the hot tub is. He measures the diameter of the top and finds that the hot tub is 6 feet across. What is the height , or depth, of the hot tub?
::Gregory的家人刚为他们的湖屋买了一个热水浴缸。热水浴缸公司告诉他的家人,浴缸里有125立方英尺的水。Gregory想知道热水浴缸有多深。他测量了顶部的直径,发现热水浴缸宽6英尺。热水浴缸的高度或深度是多少?

In this concept, you will learn how to calculate the height of a cylinder when given the and radius or diameter.
::在此概念中, 当给定圆柱体的高度、半径或直径时, 您将学会如何计算圆柱体的高度。

Finding the Height of a Cylinder Given Volume
::寻找圆柱体给定量的高度

Sometimes you will know the volume and radius of a cylinder and you won’t know the height of it. Think about a water tower that is cylindrical in shape. You might know how much volume the tank will hold and the radius of the tank, but not the height of it. When this happens, you can use the formula for the volume of a cylinder to find the missing height:
::有时你会知道一个圆柱体的体积和半径,而你也不会知道它的高度。想想一个圆柱形的水塔。你可能会知道罐体的体积和圆柱体的半径,但不知道罐体的半径。当发生这种情况时,你可以使用圆柱体体体体积的公式来找到缺失的高度:

$\begin{aligned} V & = π r^{2} h \\ V & = π (2)^{2} (10) \\ V & = π (4) (10) \\ V & = 40 π \\ V & = 125.6 i n .^{3} \end{aligned}$

::Vr2hV(2)2(10)V(4)(10)V=40V=125.6英寸。

Let's look at an example.
::让我们举个例子。

A cylinder with a radius of 2 inches has a volume of 125.6 cubic inches. What is the height of the cylinder?
::半径为2英寸的圆柱体体体积为125.6立方英寸。圆柱体的高度是多少?

The volume and the radius are given, so substitute these into the formula and then solve for $\begin{aligned} h \end{aligned}$ , the height.
::给定音量和半径, 将其替换为公式, 然后解析 h, 高度。

$\begin{aligned} V & = π r^{2} h \\ 125.6 & = (3.14) (2^{2}) h \\ 125.6 & = (3.14) (4) h \\ 125.6 & = 12.56 h \\ 125.6 \div 12.56 & = 12.56 h \div 12.56 \\ 10 i n & = h \end{aligned}$

::Vr2h125.6=(3.14)(22)(22)h125.6=(3.14)(4)h125.6=12.56h125.6*12.56=12.56*12.56*12.5610 in=h

The height of the cylinder is 10 inches.
::气瓶的高度是10英寸

Check your work by substituting the answer in for the height. You should get a volume of 125.6 cubic inches.
::检查您的工作, 将答案替换为高度。您应该得到125.6 立方英寸的体积。

$\begin{aligned} V & = π r^{2} h \\ V & = π (2)^{2} (10) \\ V & = π (4) (10) \\ V & = 40 π \\ V & = 125.6 i n .^{3} \end{aligned}$

::Vr2hV(2)2(10)V(4)(10)V=40V=125.6英寸。

What is the height of a cylinder that has a radius of 6 cm and a volume of 904.32 cubic cm?
::半径为6厘米、体积为904.32立方厘米的圆柱体的高度是多少?

Again, you have been given the volume and the radius. Put this information into the formula along with the value of pi and solve for $\begin{aligned} h \end{aligned}$ , the height.
::再次向您提供了音量和半径。将这些信息与 pi 值和 h, 高度的解析值一起放入公式中。

$\begin{aligned} V & = π r^{2} h \\ 904.32 & = (3.14) (6^{2}) h \\ 904.32 & = (3.14) (36) h \\ 904.32 & = 113.04 h \\ 904.32 \div 113.04 & = 113.04 h \div 113.04 \\ 8 c m & = h \end{aligned}$

::Vr2h904.32=(3.14)(62)(62)h904.32=(3.14)(36)h904.32=113.04 h904.32*113.04=113.04*113.04=113.04h*113.0448 cmm=h

The height of this cylinder is 8 centimeters.
::这个气瓶的高度是8厘米

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Gregory and his family's hot tub.
::早些时候,你得到一个问题关于格雷戈里和他的家人的热水浴缸。

To figure this out, use the formula for volume of a cylinder. He already knows the volume of the tub is 125 cubic feet and the diameter is 6 feet.
::他已经知道浴缸的体积是125立方英尺,直径是6英尺。

First, divide the diameter by 2 and plug the values for volume, pi, and radius into the formula for volume of a cylinder.
$\begin{aligned} r & = 6 \div 2 \\ r & = 3 \end{aligned}$

$\begin{aligned} V & = π r^{2} h \\ 125 & = (3.14) (3^{2}) h \end{aligned}$

::首先,将直径除以 2 ,并将体积、 pi 和半径的值插入气瓶.r = 6 2r = 3 Vr2h125= (3.14)(32) 体积的公式中。

Next, square the radius and multiply the values together.
$\begin{aligned} 125 & = (3.14) (3^{2}) h \\ 125 & = (3.14) (9) h \end{aligned}$

::下一个, 方形半径, 乘以数值。 125=( 3. 14)( 32)( 32)( 32) h125=( 3. 14)( 9) h

$\begin{aligned} 125 & = 28.26 h \end{aligned}$

::125=28.26小时

Last, divide both sides by 200.96 for the answer, remembering to include the appropriate unit of measurement.
::最后,将双方除以200.96作为答案,要记住包括适当的计量单位。

$\begin{aligned} 125 & = 28.26 h \\ 125 \div 28.26 & = 28.26 h \div 28.26 \\ 4.42 f t & = h \end{aligned}$

::125=28.26h125 28.26=28.26h 28.26.42英尺=h

The answer is Gregory's hot tub is 4.42 feet deep.
::答案是Gregory的热浴缸深4.42英尺

Example 2
::例2

Javier wants to construct a cylindrical container to hold enough water for his pet fish. He read that the fish needs to live in 2,110.08 cubic inches of water. If he constructs a tank with a diameter of 16 inches, how tall must he make it so that it holds the right amount of water?
::哈维尔想建造一个圆柱形容器,为他的宠物鱼提供足够水。他读到鱼需要生活在2,110.08立方英寸的水中。如果他建造一个直径为16英寸的坦克,那么它必须有多高才能保持适当的水量?

First, divide the diameter by 2 and plug the values for volume, pi, and radius into the formula for volume of a cylinder.
::首先,将直径除以 2,并将体积、 pi 和半径的值插入气瓶体积的公式中。

$\begin{aligned} r & = 16 \div 2 \\ r & = 8 \end{aligned}$

::r = 16 2r = 8

$\begin{aligned} V & = π r^{2} h \\ 2, 110.08 & = (3.14) (8^{2}) h \end{aligned}$

::Vr2h2,110.08=(3.14)(82)h

Next, square the radius and multiply the values together.
::下一步,将半径平方,然后将值乘在一起。

$\begin{aligned} 2, 110.08 & = (3.14) (8^{2}) h \\ 2, 110.08 & = (3.14) (64) h \end{aligned}$

::2,110.08=(3.14)(82)h2,110.08=(3.14)(64)h

$\begin{aligned} 2, 110.08 & = 200.96 h \end{aligned}$

::2,110.08=200.96h

Then, divide both sides by 200.96 for the answer, remembering to include the appropriate unit of measurement.
::然后将双方除以200.96来回答问题,记住要包括适当的计量单位。

$\begin{aligned} 2, 110.08 & = 200.96 h \\ 2, 110.08 \div 200.96 & = 200.96 h \div 200.96 \\ 10.5 i n & = h \end{aligned}$

::2,110.08=200.96h2,110.08200.96=200.96h200.96.10.5英寸h

The answer is Javier must make his tank 10.5 inches tall for his tank to hold 2,110.08 cubic inches of water.
::答案是哈维尔必须把他的坦克10.5英寸高才能保持2110.08立方英寸的水

Example 3
::例3

Find the height of a cylinder with radius = 6 inches and volume = 904.32 cubic inches.
::查找圆柱体的高度,半径=6英寸,体积=904.32立方英寸。

First, plug the values of the volume, pi, and radius into the formula for volume of a cylinder.
::首先,将体积、 pi 和半径的值插入气瓶体积的公式中。

$\begin{aligned} V & = π r^{2} h \\ 904.32 & = (3.14) (6^{2}) h \end{aligned}$
Next, square the radius and multiply the values together.
::V r2h904.32=(3.14)(62)(62)h 下一步,将半径平方,将值相乘。

$\begin{aligned} 904.32 & = (3.14) (6^{2}) h \\ 904.32 & = (3.14) (36) h \end{aligned}$

::904.32=(3.14)(62)h904.32=(3.14)(36)h

$\begin{aligned} 904.32 & = 113.04 h \end{aligned}$

::904.32=113.04h

Last, divide each side by 113.04 for the answer, remembering to include the appropriate unit of measurement.
::最后,将每一方的答案除以113.04,记住包括适当的计量单位。

$\begin{aligned} 904.32 & = 113.04 h \\ 904.32 \div 113.04 & = 113.04 h \div 113.04 \\ 8 i n & = h \end{aligned}$

::904.32=113.04h904.32113.04=113.04h113.0448英吋

The answer is the height of the cylinder is 8 inches.
::答案是圆筒的高度是8英寸

Example 4
::例4

Find the height of a cylinder with radius = 3 meters and volume = 354.34 cubic meters.
::找到圆柱体的高度,半径=3米,体积=354.34立方米。

First, plug the values of the volume, pi, and radius into the formula for volume of a cylinder.
::首先,将体积、 pi 和半径的值插入气瓶体积的公式中。

$\begin{aligned} V & = π r^{2} h \\ 354.34 & = (3.14) (3^{2}) h \end{aligned}$
Next, square the radius and multiply the values together.
::V r2h354.34=(3.14)(32)h 下一个,半径平方,数值相乘。

$\begin{aligned} 354.34 & = (3.14) (3^{2}) h \\ 904.32 & = (3.14) (9) h \end{aligned}$

::354.34=(3.14)(32)h904.32=(3.14)(9)h

$\begin{aligned} 354.34 & = 28.26 h \end{aligned}$

::354.34=28.26h

Last, divide each side by 28.26 for the answer, remembering to include the appropriate unit of measurement.
::最后,将每一方除以28.26作为答案,要记住包括适当的计量单位。

$\begin{aligned} 354.34 & = 28.26 h \\ 354.34 \div 28.26 & = 28.26 h \div 28.26 \\ 9 m & = h \end{aligned}$

::354.34=28.26h354.3428.26=28.26h28.2669米=h

The answer is the height of the cylinder is 9 meters.
::答案是圆柱体的高度是9米

Example 5
::例5

Find the height of a cylinder with radius = 5 feet and volume = 785 cubic feet.
::找到圆柱体的高度,半径=5英尺,体积=785立方英尺。

First, plug the values of the volume, pi, and radius into the formula for volume of a cylinder.
::首先,将体积、 pi 和半径的值插入气瓶体积的公式中。

$\begin{aligned} V & = π r^{2} h \\ 785 & = (3.14) (5^{2}) h \end{aligned}$

::V r2h785=(3.14)(52)h

Next, square the radius and multiply the values together.
$\begin{aligned} 785 & = (3.14) (5^{2}) h \\ 785 & = (3.14) (25) h \end{aligned}$

::下一个,方形半径,乘以数值,共785=(3.14)(52)(52)h785=(3.14)(25)h

$\begin{aligned} 785 & = 78.5 h \end{aligned}$

::785=78.5小时

Last, divide each side by 78.5 for the answer, remembering to include the appropriate unit of measurement.
::最后,每一方除以78.5作为答案,要记住包括适当的计量单位。

$\begin{aligned} 785 & = 78.5 h \\ 785 \div 78.5 & = 78.5 \div 78.5 \\ 10 f t & = h \end{aligned}$

::785=78.5h78578.5=78.578.578.510平方英尺=h

The answer is the height of the cylinder is 10 feet.
::答案是圆柱体的高度是10英尺

Review
::回顾

Given the volume and the radius, find the height of each cylinder. Use 3.14 for the value $\begin{aligned} π \end{aligned}$ and round your answer to the nearest tenths place where needed.
::给定音量和半径, 找到每个气瓶的高度。用 3. 14 表示 ° 的值, 并按需要将您的回答绕到最近的十分之一位置。

$\begin{aligned} r = 6 i n, V = 904.32 i n^{3} \end{aligned}$
::r=6, V=904.32, 3英寸

$\begin{aligned} r = 5 i n, V = 706.5 i n^{3} \end{aligned}$
::r=5 英寸, V=706.5 英寸3

$\begin{aligned} r = 7 f t, V = 2307.9 f t^{3} \end{aligned}$
::r=7英尺,V=2307.9英尺

$\begin{aligned} r = 8 f t, V = 4019.2 f t^{3} \end{aligned}$
::r=8英尺,V=4019.2英尺

$\begin{aligned} r = 7 f t, V = 1538.6 f t^{3} \end{aligned}$
::r=7英尺,V=1538.6英尺

$\begin{aligned} r = 12 m, V = 6330.24 m^{3} \end{aligned}$
::r=12米,V=6330.24立方米

$\begin{aligned} r = 9 m, V = 4069.49 m^{3} \end{aligned}$
::r=9米,V=4069.49立方米

$\begin{aligned} r = 10 m, V = 5652 m^{3} \end{aligned}$
::r=10米,V=5652立方米

$\begin{aligned} r = 12 i n, V = 11304 i n^{3} \end{aligned}$
::r=12英寸,V=11304英寸3

$\begin{aligned} r = 11 f t, V = 3039.52 f t^{3} \end{aligned}$
::r=11英尺,V=3039.52英尺

$\begin{aligned} r = 10 i n, V = 1570 i n^{3} \end{aligned}$
::r=10英寸,V=1 570英寸3

$\begin{aligned} r = 9.5 i n, V = 1700.31 i n^{3} \end{aligned}$
::r=9.5英寸,V=1700.31英寸,3英寸

$\begin{aligned} r = 8 m, V = 1808.64 m^{3} \end{aligned}$
::r=8米,V=1808.64立方米

$\begin{aligned} r = 14 f t, V = 5538.96 f t^{3} \end{aligned}$
::r=14英尺,V=5538.96英尺

$\begin{aligned} r = 13.5 i n, V = 4005.85 i n^{3} \end{aligned}$
::r=13.5英寸, V=4005.85英寸, 3英寸

Review (Answers)
::回顾(答复)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

Resources
::资源

章节大纲

常规

气瓶体积高地

Finding the Height of a Cylinder Given Volume ::寻找圆柱体给定量的高度

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Resources ::资源

Finding the Height of a Cylinder Given Volume
::寻找圆柱体给定量的高度

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)

Resources
::资源