节: 弧的长度 | 5.4 弧的长度

章节大纲

Introduction
::导言

The central pivot irrigation system provides moisture for crops to a portion of a field by slowly rotating a long pipe that sprays water onto the land. The shape of the area irrigated is a portion of a circle. Its radius is the length of the main pipe that is slowly rotated. The grower needs to know the length of the circular arc that is drawn by the end of the irrigation pipe . For example, some days the pipe is only turned through an angle of $\begin{align*}54^{\circ}\end{align*}$ . If the pipe is 400 ft long, what is the length of the arc traced by the end of the pipe?
::中央支流灌溉系统通过缓慢旋转向土地喷洒水的长管,为田地的一部分作物提供湿度。灌溉区的形状是圆的一部分。其半径是旋转缓慢的主管长度。种植者需要知道灌溉管道尽头所绘制圆弧的长度。例如, 有几天, 管道仅经过54英寸角转动。如果管道长400英尺, 管道尽头所追踪的弧长度是多少?

Arc Length
::弧弧长度

The length of an arc on a circle depends on both the angle of rotation and the radius length of the circle. The circumference of a circle $\begin{align*}C=2\pi{r}\end{align*}$ is related to arc length. This is the arc length for one rotation of $\begin{align*}360^{\circ}\end{align*}$ or $\begin{align*}2\pi\end{align*}$ radians. I f the radius is 4 cm, then the length of the half-circle arc would be $\begin{align*}4\pi\end{align*}$ cm.
::圆的弧长度取决于圆的旋转角度和圆的半径长度。圆 C=2°r 的环绕与弧长度有关。这是一个旋转360° 或 2弧度的弧长度。如果半径为 4 厘米,则半圈弧的长度为 4 厘米。

Extending this idea to an arbitrary angle $\begin{align*}\theta\end{align*}$ inside the circle results in a formula t hat can be used to calculate the length of any arc.
::将这一想法扩展至圆圈内的任意角度 , 产生一个公式, 可用于计算任何弧的长度。

Arc Length
::弧弧长度

$\begin{aligned} s = r θ, \end{aligned}$
$\begin{align*}s = r \theta,\end{align*}$
::s=r,

where $\begin{align*}s\end{align*}$ is the length of the arc, $\begin{align*}r\end{align*}$ is the radius, and $\begin{align*}\theta\end{align*}$ is the measure of the angle in radians.
::s 是弧的长度, r 是半径, \ 是以弧度衡量角。

Examples
::实例

Example 1
::例1

The free-throw line on an NCAA basketball court is 12 ft wide. In international competition, it is only about 11.81 ft. How much longer is the half circle above the free-throw line on the NCAA court?
::NCAA篮球场的自由发射线宽12英尺,在国际竞争中,只有11.81英尺。 NCAA法院的自由发射线上方的半圈还有多久?

Solution:
::解决方案 :

Find both arc lengths.
::查找两个弧长度。

$\begin{aligned} NCAA & INTERNATIONAL \\ s_{1} = r θ & s_{2} = r θ \\ s_{1} = \frac{12}{2} (π) & s_{2} \approx \frac{11.81}{2} (π) \\ s_{1} = 6 π & s_{2} \approx 5.905 π \end{aligned}$
$\begin{align*}& \text{NCAA} && \text{INTERNATIONAL}\\ & s_1=r \theta && s_2 =r \theta\\ & s_1=\frac{12}{2}(\pi) && s_2 \approx \frac{11.81}{2}(\pi)\\ & s_1=6\pi && s_2 \approx 5.905\pi\end{align*}$
::NCAINTIONs1=rQs2=rQs1=122()s211.812()s1=6Qs25.905}

So the answer is approximately $\begin{align*}6 \pi - 5.905 \pi \approx 0.095\pi\end{align*}$ .

::答案大约是5905 0.095 。

This is approximately 0.3 ft, or about 3.6 in longer.
::大约为0.3英尺,或3.6英尺长。

Example 2
::例2

Two connected gears are rotating. The smaller gear has a radius of 4 in and the larger gear’s radius is 7 in. What is the angle through which the larger gear has rotated when the smaller gear has made one complete rotation?
::两个连接的齿轮正在旋转。小齿轮的半径为4, 大齿轮的半径为7。当较小的齿轮完全旋转时,较大的齿轮的旋转角度是什么?

Solution:
::解决方案 :

Step 1: Because the blue gear performs one complete rotation, the length of the arc traveled is calculated using the blue gear radius and the circumference of a whole circle:
::第1步:由于蓝色渔具进行一个完整的旋转,所穿行弧的长度是使用蓝色渔具半径和整个圆环来计算的:

$\begin{aligned} s & = r θ \\ s & = 4 \times 2 π \\ s & = 8 π \end{aligned}$
$\begin{align*}s &= r\theta\\ s&=4 \times 2 \pi \\ s&=8 \pi\end{align*}$
::s=rs=4×2s=8

Step 2: Use the $\begin{align*}8\pi\end{align*}$ arc length and the radius of the larger gear to determine the angle formed on the larger circle:
::第2步:使用大渔具的8弧长度和半径来确定大圆上形成的角度:

$\begin{aligned} θ & = \frac{s}{r} \\ θ & = \frac{8 π}{7} \\ θ & \approx 3.6 radians \end{aligned}$
$\begin{align*}\theta &=\frac{s}{r}\\ \theta &=\frac{8\pi}{7}\\ \theta &\approx 3.6 \text{ radians}\end{align*}$
::873.6弧度

Step 3: The angle is approximately 3.6 radians. Convert this answer to degrees using the fact that $\begin{align*}\pi\end{align*}$ is equal to $\begin{align*}180^{\circ}\end{align*}$ .
::第3步:角度约为3.6弧度。使用等于 180 的这一事实将答案转换为度。

$\begin{aligned} 3.6 \times \frac{180^{\circ}}{π} \approx 206^{\circ} \end{aligned}$
$\begin{align*}3.6 \times \frac{180^{\circ}}{\pi} \approx 206^{\circ}\end{align*}$

Example 3
::例3

The radius of a standard car tire is 27.94 cm. How far does a car go in one revolution of the tire?
::标准车胎的半径是27.94厘米。汽车在轮胎的一次革命中走多远?

Solution:
::解决方案 :

Since the distance traveled by the tire is equal to the distance around the tire, we can use the circumference of the tire to answer the question:
::因为轮胎行驶的距离相当于轮胎周围的距离, 我们可以用轮胎的环绕来回答问题:

$\begin{aligned} s & = r θ \\ s & = (27.94 cm) (2 π) \\ s & = 175.55 cm \end{aligned}$
$\begin{align*}s &= r\theta\\ s &= (27.94 \text{ cm})(2\pi)\\ s &= 175.55 \text{ cm} \end{align*}$
::=rs=(27.94厘米)(2)s=175.55厘米

Example 4
::例4

To return to the problem in the Introduction, the pipe is slowly rotated through an angle of $\begin{align*}54^{\circ}.\end{align*}$ If the pipe is 400 ft long , calculate the length of the arc traced by the end of the irrigation pipe.

::回到引言中的问题, 管道会慢慢旋转到54英寸角。如果管道长400英尺, 请计算灌溉管道尾端所追踪的弧长度。

Solution:
::解决方案 :

Step 1: Convert the angle from degrees to radians.
::第1步:将角度从度转换为弧度。

$\begin{aligned} θ = 54^{\circ} \cdot \frac{π}{180^{\circ}} = 0.3 π = 0.942 radians \end{aligned}$
$\begin{align*}\theta = 54^{\circ} \cdot \frac{\pi}{180^{\circ}}=0.3 \pi = 0.942 \text{ radians} \end{align*}$
::541800.30.942弧度

Step 2: Calculate the arc length.
::第2步:计算弧长度。

$\begin{aligned} s = 400 \cdot 0.3 π radians = 120 π \approx 377 ft \end{aligned}$
$\begin{align*}s= 400 \cdot 0.3\pi \text{ radians}=120 \pi \approx 377 \text { ft}\end{align*}$
::s=4000.3弧度=120377平方英尺

Example 5
::例5

A stranded motorist is trying to push a car after it has broken down. If the car simply rocks back and forth rather than move forward, how far did the tire move each time the car rocked? If the radius of the car's tire is 14 in, and the change in the tire's angle is $\begin{align*}\frac{\pi}{2}\end{align*}$ radians, how far did the tire move?
::一个停靠的驾驶员试图在汽车破裂后推车。如果汽车只是来回摇动,而不是向前移动,那么每次车震动时轮胎移动多远?如果车胎半径是14英寸,轮胎角度的变化是+2弧度,轮胎移动多远?

Solution:
::解决方案 :

Since the distance the tire moved is equal to the length of the arc the tire rolled, use $\begin{align*}s = r\theta\end{align*}$ to determine how far the tire went:
::由于轮胎移动的距离等于轮胎滚动弧的长度,使用S=r确定轮胎走多远:

$\begin{aligned} s & = r θ \\ s & = (14) (\frac{π}{2}) \\ s & = 7 π \\ s & \approx 21.99 i n \end{aligned}$
$\begin{align*}s &= r\theta\\ s &= (14)(\frac{\pi}{2})\\ s &= 7\pi\\ s &\approx 21.99 in\\ \end{align*}$
::s=rs=( 14) (%2)s= 7s=21. 99in

Example 6
::例6

If an object with a radius of 10 cm spins so that its arc covers 54 cm, what is the change in angle of the object?
::如果一个半径为 10 厘米旋转的物体使其弧覆盖54 厘米,则对象角度的变化是什么?

Solution:
::解决方案 :

Use the equation $\begin{align*}s = r\theta\end{align*}$ to solve this problem:
::使用方程式 s=r 来解决这个问题 :

$\begin{aligned} θ = \frac{s}{r} = \frac{54}{10} = 5.4 \end{aligned}$
$\begin{align*}\theta = \frac{s}{r} = \frac{54}{10} = 5.4\\ \end{align*}$
::@sr=5410=5.4

The disk moves 5.4 radians, which is a little less than a complete rotation, since a complete rotation is approximately 6.28 radians.
::磁盘移动5.4弧度,略低于完全旋转,因为完全旋转约为6.28弧度。

Example 7
::例7

A DVD has a radius of 2.25 in, how far does a point on the disk spin if the player turns it $\begin{align*}\frac{\pi}{2}\end{align*}$ radians?
::DVD半径为 2. 25, 如果播放器旋转 2 弧度, 磁盘旋转点有多远 ?

Solution:
::解决方案 :

Using $\begin{align*}s = r\theta\end{align*}$ ,
::使用 s=r,

$\begin{aligned} s = (2.25) (\frac{π}{2}) \\ s = 1.125 π \approx 3.534 \end{aligned}$
$\begin{align*}s = (2.25)(\frac{\pi}{2})\\ s = 1.125\pi \approx 3.534\\ \end{align*}$
::s=( 2.25)( 2)s=1.125 3.534

A point on the disk turns 3.534 in.
::磁盘上的一个点翻转3.534英寸。

Summary
::摘要

The length of any arc is $\begin{align*}s = r \theta,\end{align*}$ where $\begin{align*}s\end{align*}$ is the length of the arc, $\begin{align*}r\end{align*}$ is the radius, and $\begin{align*}\theta\end{align*}$ is the measure of the angle in radians.
::任何弧的长度为 s=r, 其中 s 是弧的长度, r 是半径, 是角以弧度的量度。

::任何弧的长度为 s=r, 其中 s 是弧的长度, r 是半径, 是角以弧度的量度。

Use the fact that $\begin{align*}\pi\end{align*}$ is equal to $\begin{align*}180^{\circ}\end{align*}$ to convert between degrees and radians.
::使用 = 等于 180 = 来转换度和弧度。

A subtended arc is the part of the circle in between the two rays that make the central angle.
::子弧是形成中心角的两个射线之间的圆圈部分。

Review
::回顾

The radius of a carousel is 8 meters. Assume you are on the outside edge. Use this information to answer questions 1-3.
::旋转木马的半径是8米。假设您在外部边缘。使用此信息回答问题 1-3 。

You move half way around the carousel. How far did you travel?
::你绕着旋转木马走过半个路程你走多远?

You move all the way around the carousel. How far did you travel?
::你绕着旋转木马到处乱跑你走多远?

You have now traveled all the way around the carousel twice. How far did you travel?
::你们已经绕着旋转木马绕了两次路了你们跑到哪里去了?

A pizza has a radius of 10 in. Use this information to answer questions 4-5.
::披萨的半径是10英寸用这个信息回答问题4 -5

A slice is removed. The length of the crust of the missing slice is 3 in. What is the central angle of the missing slice?
::切片被切除。遗漏切片的地壳长度是 3 英寸。遗漏切片的中心角度是什么 ?

You eat three pieces with a total central angle of $\begin{align*}\frac{4\pi}{5}\end{align*}$ . What is the length of the crust you ate?
::你吃了三片中央角度是4 5,你吃了多少地壳的长度?

A large pizza has a radius of 12 in. What is the length of the crust of half the large pizza?
::大披萨的半径是12英寸大披萨的一半的地壳长度是多少?

The diameter of a tire is 35 in. Use this information to answer questions 7-10.
::轮胎直径为35英寸,使用这些信息回答问题7-10。

What is the length around the whole tire?
::整个轮胎的长度是多少?

The tire travels one mile (5280 ft). How many revolutions did the tire make?
::轮胎行驶1英里(5280英尺)。轮胎进行了多少次革命?

You roll the tire so it rotates $\begin{align*}7\pi\end{align*}$ radians. How far did it move?
::你把轮胎卷起来,然后它旋转7弧度它移动了多远?

The tire travels half a mile. How many radians did the tire rotate?
::轮胎转动了多少弧度?

Consider a standard 12-hour clock like the one below with a radius of 5 in. Use this to answer questions 11-15.
::考虑一个标准的12小时钟表, 如下面的时钟, 半径为 5 。用它回答问题 11-15 。

What is the length of the arc between the 3 and the 7?
::3和7之间的弧长度是多少?

What is the length of the arc between the 3 and the 2?
::3和2之间的弧长度是多少?

It is 12:30. What is the length of the arc between the minute and hour hands?
::现在是12: 30 分钟与小时之间方弧的长度是多少?

It is 7:20. What is the length of the arc between the minute and hour hands?
::现在是7: 20。分钟与小时之间的弧长度是多少?

It is 1:25. What is the length of the arc between the minute and hour hands?
::现在是1: 25 。分钟与小时手之间的弧长度是多少?

Review ( Answers )
::回顾(答复)

Please see the Appendix.
::请参看附录。

章节大纲

Introduction ::导言

Arc Length ::弧弧长度

Arc Length ::弧弧长度

Examples ::实例

Summary ::摘要

Review ::回顾

Review ( Answers ) ::回顾(答复)

Introduction
::导言

Arc Length
::弧弧长度

Arc Length
::弧弧长度

Examples
::实例

Summary
::摘要

Review
::回顾

Review ( Answers )
::回顾(答复)