Course: 5.11缩放距离或尺寸

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 缩放距离或尺寸

Collapse Expand
缩放距离或尺寸
Sebastian, Abe, and Rajeesh have taken up geocaching. The boys have use of a GPS and can see the general location of the cache on a map. If the scale is 1" = 80 , and the map shows the cache as being 2 inches from the boys' current location, how far do they have to walk to reach it?
::Sebastian, Abe, and Rajeesh 已经进入了地理剖析。男孩们使用GPS, 可以在地图上看到缓存的大致位置。如果比例尺是1” = 80, 地图显示缓存距离男孩们目前的位置2英寸, 他们需要走多远才能到达?

In this concept, you will learn about scale distances.
::在这个概念中,你会了解尺度距离。

Scale Distances or Dimensions
::缩放距离或尺寸

A ratio compares two quantities. Ratios can be written as fractions, with a colon or with the word “to.”
::A 比率比较两个数量。比率可以以分数表示, 用冒号或“ to”字表示。

$\begin{aligned} \frac{2}{3} \end{aligned}$ , 2:3, and "2 to 3" are ratios.
::2,3,2,3,2,3 和"2,3"是比例。

A proportion is created when two ratios are found to be equivalent or equal.
::当发现两个比率对等或相等时,就形成一个比例。

$\begin{aligned} \frac{1}{2} = \frac{3}{6} \end{aligned}$ is a proportion.
::1 2 = 3 6 是一个比例。

A unit rate is a comparison of two measurements, one of which has a value of 1.
::A单位费率是两项测量的比较,其中一项测量值为1。

55 miles per hour , $\begin{aligned} \frac{55}{1} \frac{m i l e s}{h o u r} \end{aligned}$ is a unit rate.
::每小时55英里 551米 i e s h o u r 是单位费率。

Somewhat like a unit rate, a unit scale is a ratio that compares two measurements.
::与单位比率类似,单位比额表是比较两种测量的比重。

A unit scale is a ratio that compares the dimensions of an actual object to the dimensions of a scale drawing or model that represents the actual object. Neither value in a unit scale has to equal 1.
::单位比例表是指将实际物体的尺寸与代表实际物体的尺度图示或模型的尺寸进行比较的比率。单位比例表中的数值不一定等于1。

The unit scale on a map may read 1 inch = 100 feet.
::地图上的单位比例尺可读取 1 英寸 = 100 英尺。

The ratio would be written $\begin{aligned} \frac{1 i n c h}{100 f t} \end{aligned}$ .
::比率为1 i n c h 100 f t。

To represent a line 500 feet long, the unit scale would be used to draw a line 5 inches long. A line 8 inches long would represent an actual line of 800 feet.
::代表一条500英尺长的线,单位比例尺将用来绘制一条5英寸长的线,8英寸长的线将代表一条800英尺的实际线。

Unit scales and proportions can be used to calculate actual distances from maps, drawings, or models. Actual distances can be represented on maps, drawings, or models by using unit scales.
::单位比例和比例可用来计算与地图、图纸或模型的实际距离,实际距离可以用单位比例表示在地图、图纸或模型上。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about the three boys who went geocaching.
::早些时候,你得到一个问题关于3个男孩谁去地理焦距。

The scale on their GPS unit is 2.5 cm = 80 meters, and the map shows the cache as being 5 cm from the boys' current location, how far do they have to walk to reach it?
::他们的GPS单位的规模是2.5厘米 = 80米,地图显示缓存距离男孩们目前的位置是5厘米,他们需要步行多远才能到达?

First, write the unit scale as a ratio $\begin{aligned} \frac{2.5 c m}{80 m e t e r s} \end{aligned}$
::首先,将单位比额表以2.5 cm 80 m e t e r s的比率写成。

$\begin{aligned} \frac{c m}{m} = \frac{2.5}{80} \end{aligned}$

Next, keeping the units consistent, write a ratio that compares the scale distance to the actual distance from the cache.
::接下来,保持单位的一致性, 写一个比例比, 将比例尺距离与缓存的实际距离作比较。

$\begin{aligned} \frac{s c a l e}{a c t u a l} i n \frac{c m}{m} = \frac{5}{d} \end{aligned}$

Next, set this ratio equal to the unit scale to form a proportion.
::下一步,设定该比率等于单位规模,以形成一个比例。

$\begin{aligned} \frac{2.5}{80} = \frac{5}{d} \end{aligned}$

Then, cross multiply.
::然后,交叉倍增。

$\begin{aligned} 2.5 d & = 400 \\ d & = 160 m e t e r s \end{aligned}$

The answer is d = 160 meters. The cache is 160 meters away.
::答案是d = 160米暗藏处在160米之外

Example 2
::例2

Rano has a rectangular backyard. Its actual dimensions are 50 feet by 30 feet. Rano wants to make a scale drawing of his backyard and has chose a scale of $\begin{aligned} \frac{1}{2} i n . = 5 f t \end{aligned}$ .
::Rano有一个长方形后院,其实际尺寸为50英尺乘30英尺。Rano想在后院上绘制一个比例图,并选择了一个比例尺为12 i n.= 5 f t。

Find the dimensions of the backyard for the scale drawing.
::找到后院的尺寸来绘制比例尺

First, write the unit scale as a ratio.
::首先,将单位比例表写成比例。

$\begin{aligned} \frac{1}{2} i n . = 5 f t \end{aligned}$

$\begin{aligned} \frac{i n c h e s}{f e e t} = \frac{0.5}{5} \end{aligned}$

Next, keeping the units consistent, write ratios that compare the scale distance to the actual distance for both the length and width of the backyard.
::其次,保持单位的一致,写法比率,将后院的长度和宽度与规模距离与实际距离进行比较。

$\begin{aligned} L e n g t h = \frac{s c a l e}{a c t u a l} i n \frac{i n c h e s}{f e e t} = \frac{1}{50} \end{aligned}$

$\begin{aligned} W i d t h = \frac{s c a l e}{a c t u a l} i n \frac{i n c h e s}{f e e t} = \frac{w}{30} \end{aligned}$

Then, set each of these ratios equal to the unit scale to form two proportions, one for length and one for width.
::然后,将每一比率设定为与单位比例表相等的单位比例,组成两个比例,一个长度,一个宽度。

$\begin{aligned} L e n g t h = \frac{l}{50} = \frac{0.5}{5} \end{aligned}$

$\begin{aligned} L e n g t h = \frac{w}{30} = \frac{0.5}{5} \end{aligned}$

Next, solve each proportion by cross multiplication.
::下一步,通过交叉乘法解决每个比例。

$\begin{aligned} L e n g t h \\ 5 l & = 0.5 \times 50 \\ 5 l & = 25 \\ l & = 5 i n c h e s \end{aligned}$

$\begin{aligned} W i d t h \\ 5 w & = 0.5 \times 30 \\ 5 w & = 15 \\ w & = 3 i n c h e s \end{aligned}$

The answer is $\begin{aligned} l = 5 i n c h e s \end{aligned}$ and $\begin{aligned} w = 3 i n c h e s \end{aligned}$ .
::答案是 l = 5 i n c h e s 和 w = 3 i n c h e s 。

Rano's scale drawing of his backyard should be 5 inches long and 3 inches wide.
::拉诺在后院的平面图画应该长5英寸宽3英寸

Example 3
::例3

A rectangular fish pond is 10 feet by 20 feet. Using the scale $\begin{aligned} \frac{1}{2} \end{aligned}$ in = 5 ft , determine the dimensions for a scale model to be used in a miniature display.
::矩形鱼池为10英尺乘20英尺。使用 12 尺= 5 英尺,确定用于微型显示器的尺度模型的尺寸。

First, write the unit scale as a ratio. $\begin{aligned} \frac{1}{2} \end{aligned}$ in = 5 ft or $\begin{aligned} \frac{1}{2} \end{aligned}$ inches per 5 feet.
::首先,将单位比例表写成一个比率。 12英寸=5英尺或每5英尺12英寸。

$\begin{aligned} \frac{i n c h e s}{f e e t} = \frac{0.5}{5} \end{aligned}$

Next, keeping the units consistent, write ratios that compare the scale distance to the actual distance for both the length and width of the pond.
::其次,保持单位的一致性,写法比率,将规模距离与池塘长度和宽度的实际距离进行比较。

$\begin{aligned} L e n g t h = \frac{s c a l e}{a c t u a l} i n \frac{i n c h e s}{f e e t} = \frac{1}{20} \end{aligned}$

$\begin{aligned} W i d t h = \frac{s c a l e}{a c t u a l} i n \frac{i n c h e s}{f e e t} = \frac{w}{10} \end{aligned}$

Then, set each of these ratios equal to the unit scale to form two proportions, one for length and one for width.
::然后,将每一比率设定为与单位比例表相等的单位比例,组成两个比例,一个长度,一个宽度。

$\begin{aligned} L e n g t h = \frac{l}{20} = \frac{0.5}{5} \end{aligned}$

$\begin{aligned} W i d t h = \frac{w}{10} = \frac{0.5}{5} \end{aligned}$

Next, solve each proportion by cross multiplication.
::下一步,通过交叉乘法解决每个比例。

$\begin{aligned} L e n g t h \\ 5 l & = 0.5 \times 20 \\ 5 l & = 10 \\ l & = 2 i n c h e s \end{aligned}$

$\begin{aligned} W i d t h \\ 5 w & = 0.5 \times 10 \\ 5 w & = 5 \\ w & = 1 i n c h e s \end{aligned}$

The answer is l = 2 inches, and w = 1 inch. The miniature fish pond should be 2 inches by 1 inch.
::答案是 l = 2 英寸, w = 1 英寸。微型鱼池应该是 2 英寸乘 1 英寸。

Example 4
::例4

A drawing has a scale of 1" = 10 ft, find the actual dimensions of a building that measures 10 inches by 14 inches on the drawing.
::绘图的尺度为 1 = 10 英尺, 找到一个建筑的实际尺寸, 以10 英寸乘14 英寸测量在绘图上。

First, write the unit scale as a ratio $\begin{aligned} \frac{1 i n c h}{10 f e e t} \end{aligned}$
::首先,将单位比例表写成1 i n c h 10 f e e e t

$\begin{aligned} \frac{i n c h e s}{f e e t} = \frac{1}{10} \end{aligned}$

Next, keeping the units consistent, write ratios that compare the scale distance to the actual distance for both the length and width of the building.
::其次,保持单位的一致,将规模距离与大楼长度和宽度的实际距离进行比较。

$\begin{aligned} L e n g t h = \frac{s c a l e}{a c t u a l} i n \frac{i n c h e s}{f e e t} = \frac{14}{l} \\ W i t h = \frac{s c a l e}{a c t u a l} i n \frac{i n c h e s}{f e e t} = \frac{10}{w} \end{aligned}$

Then, set each of these ratios equal to the unit scale to form two proportions, one for length and one for width.
::然后,将每一比率设定为与单位比例表相等的单位比例,组成两个比例,一个长度,一个宽度。

$\begin{aligned} L e n g t h = \frac{14}{l} = \frac{1}{10} \\ W i d t h = \frac{10}{w} = \frac{1}{10} \end{aligned}$

Next, solve each proportion by cross multiplication.
::下一步,通过交叉乘法解决每个比例。

$\begin{aligned} L e n g t h \\ l & = 10 \times 14 \\ l & = 140 f e e t \\ W i d t h \\ w & = 10 \times 10 \\ w & = 100 f e e t \end{aligned}$

The answer is l = 140 feet, and w = 100 feet. The actual dimensions of the building are 140 ft by 100 feet.
::答案是 l = 140 英尺, w = 100 英尺。大楼的实际尺寸是 140 英尺乘 100 英尺。

Example 5
::例5

Below is a map of Camp Skyview.
::下面是Skyview营地地图。

This morning, Mia walked directly from her bunk to the dining hall for breakfast. Then she walked directly to the Art & Craft Center. How many meters did she walk altogether?
::今天早上,米娅直接从床铺走到餐厅吃早餐,然后直接走到艺术和手工艺中心。

First, add together the distances from the map that show how far Mia walked
::首先,加上地图的距离,显示米娅走多远

4 cm + 2.25 cm = 6.25 cm
::4厘米+2.25厘米=6.25厘米

Next, write the unit scale as a ratio $\begin{aligned} \frac{0.5 c m}{14 m e t e r s} \end{aligned}$
::下一步,将单位比额表写成0.5cm 14m e t e r s

Then, keeping the units consistent, write a ratio that compares the scale distance to the actual distance that Mia walked.
::然后,保持单位的一致性, 写一个比例比比较比例距离与实际距离米娅走。

$\begin{aligned} \frac{s c a l e}{a c t u a l} i n \frac{c m}{m} = \frac{6.25}{d} \end{aligned}$

Next, set this ratio equal to the unit scale to form a proportion.
::下一步,设定该比率等于单位规模,以形成一个比例。

$\begin{aligned} \frac{0.5}{14} = \frac{6.25}{d} \end{aligned}$

Then, cross multiply.
::然后,交叉倍增。

$\begin{aligned} 0.5 d & = 6.25 \times 14 \\ d & = 175 m e t e r s \end{aligned}$

The answer is d = 175 meters. Mia walked a total of 175 meters.
::答案是 d = 175米米娅共走了175米

Review
::回顾

Solve each problem using scale and measurement.
::使用规模和计量方法解决每个问题。

Haley made a scale model of her new school building. The scale she used for her model was 1 inch = 6 feet. The actual height of her school building is 30 feet. What was the height of the school building in her scale model?
::Haley为她的新校舍做了一个规模模型,模型使用的尺度为1英寸=6英尺,学校建筑的实际高度为30英尺。规模模型中的学校建筑的高度是多少?

If the width of Haley’s school is 120 feet, what would be the width in the scale model?
::如果Haley的学校宽度是120英尺,那么规模模型的宽度是多少?

If the length of Haley’s school is 180 feet, what would be the length in the scale model?
::如果Haley的学校长度为180英尺,那么规模模型的长度是多少?

Eddie drew a map of Main Street in his hometown. The scale he used for his map was 1 centimeter = 8 meters. The actual distance between the post office and City Hall, both of which are on Main Street, is 56 meters. What is the distance between those two places on Eddie's map?
::Eddie在家乡绘制了一份主街地图。他的地图上所用的比例尺是1厘米=8米。邮局和市政厅之间的实际距离是56米。在Eddie的地图上这两个地方之间的距离是多少?

If the distance from the post office to the library is twice the distance from the post office to the City Hall, what is the distance on Eddie’s map?
::如果邮局到图书馆的距离是邮局到市政厅的两倍,

If the distance from the library to the school is three times the distance as from the post office to the City Hall, what is the distance on Eddie’s map?
::如果从图书馆到学校的距离是邮局到市政厅的距离的三倍,埃迪地图上的距离是多少?

Asharah built a model of a car. The actual length of the car is 12 feet. The scale of her model is $\begin{aligned} \frac{1}{4} i n c h = 1 f o o t \end{aligned}$ . What is the length of her model car?
::Asharah建造了一辆汽车的模型。汽车的实际长度是12英尺。她的模型的尺寸是14 i n c h = 1 f o o 。她的模型车的长度是多少?

Kenya built a model of the same car. His scale of the model is $\begin{aligned} {\frac{1}{2}}^{″} = 1 f o o t \end{aligned}$ . What is the length of his model?
::肯尼亚建造了同一辆车的模型。模型的大小是12英寸=1英尺。模型的长度是多少?

Jonah observed a spider that was too small to draw at its actual size. So he made a scale drawing, using the scale 0.5 centimeter = 4 millimeters. The actual length of the spider's body, not including its legs, was 16 millimeters. What is the length of the spider's body, not including its legs, in Jonah's drawing?
::Jonah观察到一只蜘蛛太小,无法按其实际大小画画。他用0.5厘米=4毫米的尺度画画。蜘蛛身体的实际长度(不包括腿)是16毫米。在Jonah的图画中,蜘蛛身体的长度(不包括腿)是多少?

If Jonah made a drawing that is half the size of this one, what would length of the spider’s body be in the new drawing?
::如果约拿画的大小是这幅画的一半, 蜘蛛身体在新画中的长度会是多少?

Alyssa made a scale drawing of her rectangular classroom. She used the scale $\begin{aligned} \frac{1}{2} i n c h = 4 f e e t \end{aligned}$ . Her actual classroom has dimensions of 32 feet by 28 feet. What are the dimensions of her classroom in the scale drawing?
::Alyssa在她的矩形教室中绘制了比例图。她使用了比例表 1 2 i n c h = 4 f e e e t。她的实际教室的尺寸是32英尺乘28英尺。在比例图中,她的教室的尺寸是多少?

Below is a scale drawing of a circular fountain. In the scale drawing, the diameter of the fountain measures 3 centimeters. What is the actual diameter of the fountain?
::下面是圆形喷泉的平面图。在比例图中,喷泉的直径是3厘米。喷泉的实际直径是多少?

On a map, Brandon measured the straight-line distance between Los Angeles, California and San Francisco, California to be 2 inches. The scale on the map shows that $\begin{aligned} \frac{1}{4} i n c h = 43 m i l e s \end{aligned}$ . What is the actual straight-line distance between Los Angeles and San Francisco?
::在一张地图上,布兰登测量了洛杉矶、加利福尼亚和加利福尼亚旧金山之间的直线距离为2英寸。地图上的尺度显示,洛杉矶和旧金山之间的实际直线距离是14 i n c h = 43 m i e s。洛杉矶和旧金山之间的实际直线距离是多少?

A butterfly that Adriana observed was too small to draw at its actual size. So, she made this scale drawing.
::Adriana观察到的一只蝴蝶太小,无法按其实际大小绘制。所以,她画了这个比例尺。

In the drawing, the wingspan of the butterfly measures 4.5 centimeters. What was the actual wingspan of the butterfly Adriana observed?
::在绘画时,蝴蝶的翅膀宽度为4.5厘米。

Jeremy made this scale model of Taipei 101, one of the tallest buildings in the world. The scale height of his model is $\begin{aligned} 2 \frac{1}{2} \end{aligned}$ inches. Find the actual height of Taipei 101.
::Jeremy制作了台北101的模型, 是世界上最高的建筑之一。他的模型的高度是212英寸。找到台北101的实际高度。

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

缩放距离或尺寸

Scale Distances or Dimensions ::缩放距离或尺寸

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

Scale Distances or Dimensions
::缩放距离或尺寸

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