Course: 10.12 气瓶体量 | The Way To Learn

Section outline

Select section General

Collapse Expand
General

Collapse all Expand all
- Select activity 新闻通告
  
  新闻通告 Forum
Select section 气瓶量

Collapse Expand
气瓶量
Charlotte wants to know how much of her favorite morning beverage, hot chocolate, fills her favorite mug. She measures the mug and finds the height is 4 inches and the diameter of the top is 3.5 inches. What is the of hot chocolate that Charlotte's mug holds?
::夏洛特想知道她最喜欢的早上饮料热巧克力填满了她最喜欢的杯子她测量了杯子,发现身高是4英寸顶部直径是3.5英寸

In this concept, you will learn how to calculate the volume of a cylinder .
::在这个概念中,你会学会如何计算圆柱体的体积。

Finding the Volume of a Cylinder
::查找圆柱体的音量

A cylinder is a solid shape that exists in three-dimensional space . A cylinder has two faces that are circles. You do not call the side of a cylinder a face because it is curved.
::一个圆柱体是一个三维空间存在的固态形状。一个圆柱体有两张圆形的面孔。你不会因为圆柱体被弯曲而称圆柱体的侧面为圆形。

The volume of a cylinder is the measure of how much three-dimensional space it takes up or holds. Imagine a thermos. If you fill the thermos with water, the volume of the thermos is the amount of water it will hold. You measure volume in cubic units , because you are multiplying three dimensions : length, width, and height. The width of the thermos is the same as the diameter of the circular face.
::圆柱体的体积是测量它取出或持有的三维空间的大小。想象一下一个热点。如果您用水填充热点, 热点的体积就是它所蓄水的水量。您用立方体体量体积, 这是因为您正在乘以三个维度: 长度、宽度和高度。热点的宽度与圆形面的直径相同。

You know that the two bases are circular, so you are going to need to know the area of the circle to figure out how much space can be contained on top of it. A cylinder is tall, or has height, so you are going to need to know the height of the cylinder.
::您知道这两个基点是圆形的, 所以您需要知道圆圈的面积, 才能知道圆圈上面有多少空间。一个圆筒是高的, 或者高的, 所以您需要知道圆圈的高度。

First, let’s think about the area of the circular bases. To find the area of a circle, use the following formula where A = area, $\begin{aligned} π \end{aligned}$ (pi) is a constant equal to 3.14 when rounded, and r = radius which is half the diameter:
::首先,让我们想一想圆形基地的区域。要找到圆形区域,请使用以下公式:A=区域,(pi)的常数等于3.14(四舍五入),r=半径半径:

$\begin{aligned} A = π r^{2} \end{aligned}$
::阿 r2

But you also need the height "> $\begin{aligned} (h) \end{aligned}$ of the cylinder to figure into this calculation. If you put all of these parts together, you have the following formula.
::但是,您也需要圆柱体的高度才能算出计算结果。如果将这些部件全部放在一起,则有以下公式。

$\begin{aligned} V = π r^{2} h \end{aligned}$
::Vr2h

Let's look at an example.
::让我们举个例子。

Find the volume of a cylinder with a radius of 5 cm and a height of 7 cm.
::查找圆柱体的体积,半径为5厘米,高度为7厘米。

Start by substituting the values of the cylinder into the formula.
::首先将圆柱体的值替换为公式。

$\begin{aligned} V & = π r^{2} h \\ V & = (3.14) (5^{2}) (7) \\ V & = (3.14) (25) (7) \\ V & = 549.5 c m^{3} \end{aligned}$

::Vr2hV=(3.14)(52)(7)V=(3.14)(25)(7)V=549.5厘米

The answer is the volume of this cylinder is 549.5 cubic centimeters.
::答案是这个气瓶的体积是549.5立方厘米

Let's look at another example.
::让我们再举一个例子。

Find the volume of a cylinder with a diameter of 12 inches and a height of 8 inches.
::找到一个直径为12英寸、高度为8英寸的圆柱体的体积。

First, notice that we have been given the diameter and not the radius. You can divide the diameter in half which will give the radius of the circular base of the cylinder, in this case, 6 inches.
::首先,请注意我们得到了直径,而不是半径。您可以将直径除以半径,这样圆柱形圆基半径为6英寸。

Now substitute the values into the formula and solve for the volume.
::现在将值替换为公式, 并解决音量问题。

$\begin{aligned} V & = π r^{2} h \\ V & = (3.14) (36) (8) \\ V & = 904.32 i n^{3} \end{aligned}$

::Vr2hV=(3.14)(36)(8)V=904.32英寸3

The answer is the volume of the cylinder is 904.32 cubic inches.
::答案是圆柱体的体积是904.32立方英寸。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Charlotte's hot chocolate mug.
::早些时候,你得到一个问题关于夏洛特的热巧克力杯。

The mug is 4 inches in height with a diameter of 3.5 inches and Charlotte wants to know the volume.
::杯子身高4英寸直径3.5英寸夏洛特想知道音量

First, divide the diameter by 2 to get the radius, and plug the values for pi , the radius, and the height into the formula for volume of a cylinder.
::首先,将直径除以 2 以获得半径,并将 pi、半径和高度的值插入气瓶体积的公式中。

$\begin{aligned} r & = 3.5 \div 2 \\ r & = 1.75 \end{aligned}$

::r = 3.5 2r = 1.75

$\begin{aligned} V & = π r^{2} h \\ V & = (3.14) ({1.75}^{2}) (4) \end{aligned}$

::Vr2hV=(3.14)(1.552)(4)

Next, square the radius.
$\begin{aligned} V & = (3.14) ({1.75}^{2}) (4) \\ V & = (3.14) (3.06) (4) \end{aligned}$

::其次,半径V=(3.14)(1.552)(1.752)(4)V=(3.14)(3.06)(4)

Then, multiply the three values together for the answer, making sure to include the appropriate unit of measurement.
::然后,将答案的三个值相乘,以确保包含适当的计量单位。

$\begin{aligned} V & = (3.14) (3.06) (4) \\ V & = 38.43 {i n}^{3} \end{aligned}$

::V=(3.14)(3.06)(4)V=38.43英寸3

The answer is the volume of Charlotte's mug is 38.43 cubic inches. Her mug will hold 38.43 cubic inches of hot chocolate.
::回答是夏洛特杯子的体积是38.43立方英寸她杯子里有38.43立方英寸热巧克力

Example 2
::例2

Find the volume of a cylinder with a radius of 3 inches and height of 7 inches.
::找到圆柱体的体积,半径为3英寸,高度为7英寸。

First, plug the values for pi, the radius, and the height into the formula for volume of a cylinder.
::首先,将 pi、半径和高度的值插入气瓶体积的公式中。

$\begin{aligned} V & = π r^{2} h \\ V & = (3.14) (3^{2}) (7) \end{aligned}$

::Vr2hV=(3.14)(32)(7)

Next, square the radius.
::下一个,方形半径。

$\begin{aligned} V & = (3.14) (3^{2}) (7) \\ V & = (3.14) (9) (7) \end{aligned}$

::V=(3.14)(32)(7)(7)V=(3.14)(9)(7)

Then, multiply the three values together for the answer, making sure to include the appropriate unit of measurement.
::然后,将答案的三个值相乘,以确保包含适当的计量单位。

$\begin{aligned} V & = (3.14) (9) (7) \\ V & = 197.82 {i n}^{3} \end{aligned}$

::V=(3.14)(9)(7)(7)V=197.82英寸3

The answer is the volume of the cylinder is $\begin{aligned} 197.82 i n^{3} \end{aligned}$ .
::答案是圆柱体的体积是197.82英寸3

Example 3
::例3

Find the volume of a cylinder with radius = 2.5 mm and height = 4 mm.
::查找圆柱体的体积,半径=2.5毫米,高度=4毫米。

First, plug the values for pi, the radius, and the height into the formula for volume of a cylinder.
::首先,将 pi、半径和高度的值插入气瓶体积的公式中。

$\begin{aligned} V & = π r^{2} h \\ V & = (3.14) ({2.5}^{2}) (4) \end{aligned}$

::Vr2hV=(3.14)(2.52)(4)

Then, square the radius.
$\begin{aligned} V & = (3.14) ({2.5}^{2}) (4) \\ V & = (3.14) (6.25) (4) \end{aligned}$

::然后,半径V=(3.14)(2.52)(4)V=(3.14)(6.25)(4)

Next, multiply the three values together for the answer, making sure to include the appropriate unit of measurement.
::下一步,将答案的三个值相乘,以确保包含适当的计量单位。

$\begin{aligned} V & = (3.14) (6.25) (4) \\ V & = 78.5 {m m}^{3} \end{aligned}$

::V=(3.14)(6.25)(4)V=78.5毫米3

The answer is the volume of the cylinder is $\begin{aligned} 78.5 m m^{3} \end{aligned}$ .
::答案是圆柱体的体积是78.5毫米3。

Example 4
::例4

Find the volume of a cylinder with diameter = 14 in and height = 9 in.
::查找直径=14的圆筒体积和高度=9的圆筒体积。

First, divide the diameter by 2 to get the radius, and plug the values for pi, the radius, and the height into the formula for volume of a cylinder.
::首先,将直径除以 2 以获得半径,并将 pi、半径和高度的值插入气瓶体积的公式中。

$\begin{aligned} r & = 14 \div 2 \\ r & = 7 \end{aligned}$

::r = 14 2r = 7

$\begin{aligned} V & = π r^{2} h \\ V & = (3.14) (7^{2}) (9) \end{aligned}$

::Vr2hV=(3.14)(72)(9)

Next, square the radius.
$\begin{aligned} V & = (3.14) (7^{2}) (9) \\ V & = (3.14) (49) (9) \end{aligned}$

::下一个,半径平方。 V=(3.14)(72)(9)(9)V=(3.14)(49)(9)

Then, multiply the three values together for the answer, making sure to include the appropriate unit of measurement.
::然后,将答案的三个值相乘,以确保包含适当的计量单位。

$\begin{aligned} V & = (3.14) (49) (9) \\ V & = 1, 384.74 {i n}^{3} \end{aligned}$

::V=(3.14)(49)(9)V=1 384.74英寸3

The answer is the volume of the cylinders 1,384.74 cubic inches.
::答案是圆柱体的体积 1 384.74立方英寸。

Review
::回顾

Given the radius and height of each cylinder, find its volume. Use 3.14 for the value $\begin{aligned} π \end{aligned}$ and round your answer to the nearest hundredths place where needed.
::给每个圆柱体的半径和高度, 查找其体积。值为 ° 时使用 3. 14 , 并按需要将您的回答绕到最近的一百个位置。

$\begin{aligned} r = 5 i n, h = 8 i n \end{aligned}$
::r=5 英寸, h=8 英寸

$\begin{aligned} r = 4 i n, h = 7 i n \end{aligned}$
::r=4 英寸, h=7 英寸

$\begin{aligned} r = 3 f t, h = 5 f t \end{aligned}$
::r=3英尺,h=5英尺

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

$\begin{aligned} r = 4 c m, h = 9 c m \end{aligned}$
::r=4厘米,h=9厘米

$\begin{aligned} r = 6 m, h = 12 m \end{aligned}$
::r=6米,h=12米

$\begin{aligned} r = 7 i n, h = 14 i n \end{aligned}$
::r=7 英寸, h=14 英寸

$\begin{aligned} r = 5 m, h = 10 m \end{aligned}$
::r=5米,h=10米

$\begin{aligned} r = 2 m, h = 11 m \end{aligned}$
::r=2米,h=11米

$\begin{aligned} r = 3 c m, h = 12 c m \end{aligned}$
::r=3厘米,h=12厘米

$\begin{aligned} r = 6 c m, h = 11 c m \end{aligned}$
::r=6厘米,h=11厘米

$\begin{aligned} r = 4 m, h = 14 m \end{aligned}$
::r=4米,h=14米

$\begin{aligned} r = 13 c m, h = 26 c m \end{aligned}$
::r=13厘米,h=26厘米

$\begin{aligned} r = 8 i n, h = 14 i n \end{aligned}$
::r= 8 英寸, h= 14 英寸

$\begin{aligned} r = 4.5 c m, h = 16.5 c m \end{aligned}$
::r=4.5厘米,h=16.5厘米

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

Section outline

General

气瓶量

Finding the Volume of a Cylinder ::查找圆柱体的音量

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Review ::回顾

Review (Answers) ::回顾(答复)

Resources ::资源

Finding the Volume of a Cylinder
::查找圆柱体的音量

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Review
::回顾

Review (Answers)
::回顾(答复)

Resources
::资源