节: 和间接计量应用 | 3.12 规模和间接计量应用

章节大纲

Scale and Indirect Measurement Applications
::和间接计量应用

One place where ratios are often used is in making maps. The scale of a map describes the relationship between distances on a map and the corresponding distances on the earth's surface. These measurements are expressed as a fraction or a ratio.
::经常使用比率的一个地方是绘制地图。地图的尺度描述地图上的距离与地球表面相应的距离之间的关系。这些测量以小数或比例表示。

So far we have only written ratios as fractions, but outside of mathematics books, ratios are often written as two numbers separated by a colon (. For example, instead of $\begin{aligned} \frac{2}{3} \end{aligned}$ , we would write 2:3.
::到目前为止,我们只有作为分数的书写比率,但除了数学书籍之外,比率往往被写成两个数字,用冒号(: : ) 分开。例如,我们不写23,而是写2: 3。

Ratios written this way are used to express the relationship between a map and the area it represents. For example, a map with a scale of 1:1000 would be a map where one unit of measurement (such as a centimeter) on the map would represent 1000 of the same unit (1000 centimeters, or 10 meters) in real life.
::以这种方式书写的比例表示地图与其所代表的区域之间的关系。例如, 比例尺为1: 1000的地图将是地图, 地图上的一个测量单位( 如厘米) 代表实际生活中同一单位的1000( 1000厘米, 即10米) 。

Real-World Application: Maps
::真实世界应用程序:地图

Anne is visiting a friend in London, and is using the map below to navigate from Fleet Street to Borough Road. She is using a 1:100,000 scale map, where 1 cm on the map represents 1 km in real life. Using a ruler, she measures the distance on the map as 8.8 cm. How far is the real distance from the start of her journey to the end?
::Anne正在访问伦敦的一个朋友,并且正在使用下面的地图从舰队街到Borough路的航行。她正在使用1:100 000比例尺地图,地图上1厘米代表实际生活1公里。她用标尺测量地图上的距离为8.8厘米。从旅程开始到终点的实际距离是多远?

The scale is the ratio of distance on the map to the corresponding distance in real life. Written as a fraction, it is $\begin{aligned} \frac{1}{100000} \end{aligned}$ . We can also write an equivalent ratio for the distance Anne measures on the map and the distance in real life that she is trying to find: $\begin{aligned} \frac{8.8}{x} \end{aligned}$ . Setting these two ratios equal gives us our proportion : $\begin{aligned} \frac{1}{100000} = \frac{8.8}{x} \end{aligned}$ . Then we can cross multiply to get $\begin{aligned} x = 880000 \end{aligned}$ .
::比例尺是地图上的距离与实际生活中相应的距离之比。拼写成一个分数, 是 1100000 。我们还可以写一个等值的比数, 来表示安妮在地图上的距离测量值, 以及她试图找到的真实生活中的距离: 8. 8x。设置这两个等值的比数让我们得到比例: 1100000=8. 8x。然后我们可以乘以乘以获得 x= 8800 。

That’s how many centimeters it is from Fleet Street to Borough Road; now we need to convert to kilometers. There are 100000 cm in a km, so we have to divide our answer by 100000.
::这就是从舰队街到Borough路的多少厘米;我们现在需要转换为公里。每公里有10万厘米,因此我们必须把答案除以10万。

$\begin{aligned} \frac{880000}{100000} = 8.8. \end{aligned}$

The distance from Fleet Street to Borough Road is 8.8 km.
::从舰队街到Borough路的距离为8.8公里。

In this problem, we could have just used our intuition: the $\begin{aligned} 1 c m = 1 k m \end{aligned}$ scale tells us that any number of cm on the map is equal to the same number of km in real life. But not all maps have a scale this simple. You’ll usually need to refer to the map scale to convert between measurements on the map and distances in real life!
::在这个问题上,我们本可以仅仅使用直觉:1厘米=1公里比例表告诉我们,地图上的任何厘米数量都等于实际生活中的千米。但并非所有地图都有如此简单的比例尺。你通常需要参考地图比例尺才能在地图上的测量量与实际生活中的距离之间转换!

Real-World Application: Determining Scale
::真实世界应用:确定规模

Antonio is drawing a map of his school for a project in math. He has drawn out the following map of the school buildings and the surrounding area
::Antonio正在为数学项目绘制学校地图。他绘制了以下学校建筑和周围地区的地图。

He is trying to determine the scale of his figure. He knows that the distance from the point marked A on the baseball diamond to the point marked B on the athletics track is 183 meters. Use the dimensions marked on the drawing to determine the scale of his map.
::他正试图确定自己身材的大小。他知道从棒球钻石A标记点到运动赛道B标记点的距离是183米。他用绘画上标记的尺寸来确定地图的大小。

We know that the real-life distance is 183 m, and the scale is the ratio $\begin{aligned} \frac{distance on map}{distance in real life} \end{aligned}$ .
::我们知道,实际寿命距离是183米,比例尺是实际生活中地图距离的比例距离。

To find the distance on the map, we use Pythagoras’ Theorem : $\begin{aligned} a^{2} + b^{2} = c^{2} \end{aligned}$ , where $\begin{aligned} a \end{aligned}$ and $\begin{aligned} b \end{aligned}$ are the horizontal and vertical lengths and $\begin{aligned} c \end{aligned}$ is the diagonal between points $\begin{aligned} A \end{aligned}$ and $\begin{aligned} B \end{aligned}$ .
::为了找到地图上的距离, 我们使用 Pytagoras 的理论: a2+b2=c2, 其中a和b是水平和垂直长度, c 是 A 点与 B 点之间的对角。

$\begin{aligned} 8^{2} + 14^{2} & = c^{2} \\ 64 + 196 & = c^{2} \\ 260 & = c^{2} \\ \sqrt{260} & = c \\ 16.12 & \approx c \end{aligned}$

::82+142=c264+196=c2260=c2260=c16.12=c

So the distance on the map is about 16.12 cm. The distance in real life is 183 m, which is 18300 cm. Now we can divide:
::地图上的距离大约是16.12厘米。真实生活中的距离是183米,也就是18300厘米。现在我们可以分开:

$\begin{aligned} Scale = \frac{16.12}{18300} \approx \frac{1}{1135.23} \end{aligned}$

::比额表=16.121830011135.23

The scale of Antonio’s map is approximately 1:1100.
::Antonio地图的大小约为1:1100。

Another visual use of ratio and proportion is in scale drawings . Scale drawings (often called plans ) are used extensively by architects. The equations governing scale are the same as for maps; the scale of a drawing is the ratio $\begin{aligned} \frac{distance on diagram}{distance in real life} \end{aligned}$ .
::比例和比例的另一个直观用途是比例图。比例图(通常称为计划)被建筑师广泛使用。比例表的方程式与地图相同; 比例图的尺度是实际生活中图表距离上的比例距离。

Real-World Application: Drawing to Scale
::Real- World 应用程序: 绘制到缩放

Oscar is trying to make a scale drawing of the Titanic, which he knows was 883 ft long. He would like his drawing to be at a 1:500 scale. How many inches long does his sheet of paper need to be?
::奥斯卡试图绘制泰坦尼克号的尺码图,他知道这个尺码是883英尺长。他希望他的图画是1:500尺的尺码。他的纸页需要多少英寸长?

We can reason intuitively that since the scale is 1:500, the paper must be $\begin{aligned} \frac{883}{500} = 1.766 f e e t \end{aligned}$ long. Converting to inches means the length is $\begin{aligned} 12 (1.766) = 21.192 i n c h e s \end{aligned}$ .
::我们可以直觉地解释,由于比例尺是1:500, 纸张必须长883500=1.766英尺。转换为英寸意味着长度是12( 1. 766)=21.192英寸。

Oscar’s paper should be at least 22 inches long.
::奥斯卡的论文至少应该有22英寸长。

Example
::示例示例示例示例

Example 1
::例1

The Rose Bowl stadium in Pasadena, California measures 880 feet from north to south and 695 feet from east to west. A scale diagram of the stadium is to be made. If 1 inch represents 100 feet, what would be the dimensions of the stadium drawn on a sheet of paper? Will it fit on a standard $\begin{aligned} 8.5 \times 11 \end{aligned}$ inch sheet of paper?
::加利福尼亚州帕萨迪纳的玫瑰碗体育场从北到南有880英尺,从东到西有695英尺,将绘制体育场的规模图。如果1英寸代表100英尺,那么在纸纸上画的体育场的尺寸会是多少?它是否适合标准的8.5×11英寸纸?

Instead of using a proportion, we can simply use the following equation : (distance on diagram) = (distance in real life) $\begin{aligned} \times \end{aligned}$ (scale). (We can derive this from the fact that $\begin{aligned} scale = \frac{distance on diagram}{distance in real life} \end{aligned}$ .)
::与其使用比例,我们可以简单地使用以下方程图表上的距离) = (真实生活中的距离) × (尺度)。 (我们可以从实际生活中的尺度=图上的距离这一事实中得出这一点。 )

Plugging in, we get
::插进去,我们得到

$\begin{aligned} height on paper = 880 f e e t \times \frac{1 i n c h}{100 f e e t} = 8.8 i n c h e s \end{aligned}$
::纸张高度=880英尺x1英寸100英尺=8.8英寸

$\begin{aligned} width on paper = 695 f e e t \times \frac{1 i n c h}{100 f e e t} = 6.95 i n c h e s \end{aligned}$
::纸张上的宽度=695英尺x1英寸 100英尺=6.95英寸

The scale diagram will be $\begin{aligned} 8.8 i n \times 6.95 i n \end{aligned}$ . It will fit on a standard sheet of paper.
::比例图将为 8.8 inx6.95 英寸。它将适合标准纸页。

Review
::回顾

A restaurant serves 100 people per day and takes in $908. If the restaurant were to serve 250 people per day, how much money would it take in?
::一个餐馆每天为100人服务,908美元购买,如果该餐馆每天为250人服务,它要花多少钱?

The highest mountain in Canada is Mount Yukon. It is $\begin{aligned} \frac{298}{67} \end{aligned}$ the size of Ben Nevis, the highest peak in Scotland. Mount Elbert in Colorado is the highest peak in the Rocky Mountains. Mount Elbert is $\begin{aligned} \frac{220}{67} \end{aligned}$ the height of Ben Nevis and $\begin{aligned} \frac{11}{12} \end{aligned}$ the size of Mont Blanc in France. Mont Blanc is 4800 meters high. How high is Mount Yukon?
::加拿大最高的山峰是育空山,面积是29867,是苏格兰最高山峰,Ben 尼维斯的面积是29867,科罗拉多的Elbert山是洛基山的最高山峰,Elbert山是22067,是本尼维斯的高度,法国是1112,是勃朗峰的高度。Mont Blanc山是4800米高。育空山有多高?

At a large high school it is estimated that two out of every three students have a cell phone, and one in five of all students have a cell phone that is one year old or less. Out of the students who own a cell phone, what proportion owns a phone that is more than one year old?
::在一所大型高中,据估计每三个学生中就有两所拥有手机,而每五个学生中就有一人拥有一岁或更小的手机。在拥有一部手机的学生中,哪个比例拥有一部超过一年的手机?

For 4-6, suppose a map of Ratio City has a scale of 1:1,000,000, where 1 centimeter on the map represents 10 kilometers in real life. Use that scale to determine the real-life distances in kilometers.
::4-6,假设比例城市地图比例尺为1:1 000 000,其中1厘米代表实际寿命10公里。使用这个比例尺来决定千米的实际寿命距离。

The distance on the map between city hall and high school is 1.2cm.
::地图上市政厅和高中之间的距离是1.2厘米。

The distance on the map between city hall and the main library is 0.6cm.
::地图上市政厅和主图书馆之间的距离是0.6cm。

The distance on the map between the main library and the high school is 0.4cm.
::主图书馆和高中之间的距离是0.4cm。

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。

章节大纲

Scale and Indirect Measurement Applications ::和间接计量应用

Real-World Application: Maps ::真实世界应用程序:地图

Real-World Application: Determining Scale ::真实世界应用:确定规模

Real-World Application: Drawing to Scale ::Real- World 应用程序: 绘制到缩放

Example ::示例示例示例示例

Example 1 ::例1

Review ::回顾

Review (Answers) ::回顾(答复)

Scale and Indirect Measurement Applications
::和间接计量应用

Real-World Application: Maps
::真实世界应用程序:地图

Real-World Application: Determining Scale
::真实世界应用:确定规模

Real-World Application: Drawing to Scale
::Real- World 应用程序: 绘制到缩放

Example
::示例示例示例示例

Example 1
::例1

Review
::回顾

Review (Answers)
::回顾(答复)