5.13使用比例因子查找实际尺寸

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 使用比例因子查找实际尺寸

  Collapse Expand

  使用比例因子查找实际尺寸

  

  [Figure 1]

  播放段落

  Sam is at the county fair looking at model train displays. The smallest one is an "N" gauge, which is a scale of 1:160. If the display area is 4 feet by 8 feet, how big an area does the display represent?
  ::Sam正在县集看火车模型显示。最小的是“ N” 仪表, 比例尺为1: 160。 如果显示区域是4英尺乘8英尺, 显示区域代表多大面积 ?

  播放段落

  In this concept, you will learn how to use a scale factor to find actual dimensions.
  ::在这个概念中,你将学会如何使用一个比例因素来找到实际的维度。

  播放段落

  Using Scale Factor to Find Actual Dimensions
  ::使用缩放因数查找实际尺寸

  播放段落

  A   ratio   compares two quantities. 
  ::A比率比较了两个数量。

  播放段落

  Ratios can be written as fractions, with a colon or with the word “to.”
  ::比率可以作为分数写成,用冒号或“to”一词写成。

  播放段落

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

  播放段落

  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。

  播放段落

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

  播放段落

  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。

  播放段落

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

  播放段落

  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。

  播放段落

  The unit scale on a map may read  $\begin{array}{r}\frac{1}{2}inch=100feet\end{array}$ .
  ::地图上的单位比例尺可读取 1 2 i n c h = 100 f e e e t 。

  播放段落

  The ratio would be written   $\begin{array}{r}\frac{0.5}{100}\end{array}$   . 
  ::比率为0.5 100。

  播放段落

  To represent a line 500 feet long, the unit scale would be used to draw a line  $\begin{array}{r}2\frac{1}{2}\end{array}$   inches  long.
  ::代表500英尺长的线,单位比例尺将用来绘制212英寸长的线。

  播放段落

  A line 4 inches long would represent an actual line of 800 feet.
  ::一条4英寸长的线等于800英尺的实际线。

  播放段落

  Unit scales and proportions can be used to calculate actual distances from maps, drawings, or models.
  ::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。

  播放段落

  Actual distances can be represented on maps, drawings, or models by using unit scales.
  ::地图、图纸或模型可使用单位比例表代表实际距离。

  播放段落

  A  scale factor  is somewhat like a unit scale. It is a ratio that compares scale dimensions to actual dimension. It differs because it does not give any specific units.
  ::比例系数有点像单位比例尺。它是将规模尺寸与实际尺寸进行比较的一个比率。它不同,因为它不给任何特定单位。

  播放段落

  A scale factor for a scale drawing might be  $\begin{array}{r}\frac{1}{200}\end{array}$ , which means that the actual size is 200 times the drawing.
  ::比额表的比额系数可能为1 200,这意味着实际规模是绘图的200倍。

  播放段落

  Scale factors can be used to determine both scale and actual sizes or dimensions.
  ::比额表因素可用于确定比额表和实际大小或大小。

  播放段落

  Examples
  ::实例

  播放段落

  Example 1
  ::例1

  播放段落

  Earlier, you were given a problem about the model train displays at the fair.
  ::早些时候,你在集市的火车模型展示上 遇到一个问题。

  播放段落

  The smallest one is an "N" gauge, which is a scale of 1:160. If the display area is 4 feet by 8 feet, how big an area does the display represent?
  ::最小的是一个“ N” 仪表, 比例尺为1: 160。 如果显示区域是 4 英尺乘8 英尺, 显示区域代表多大 ?

  播放段落

  First, write the scale factor.
  ::首先,写下比额表系数。

  $$\begin{array}{r}\frac{scale}{actual}=\frac{1}{160}\end{array}$$

  播放段落

  Next, write ratios to represent the unknown actual  length  and width of the display to the scale length and width of the display.
  ::下一步,写率表示显示显示的大小长度和宽度未知的实际长度和宽度。

  $$\begin{array}{rl}Length& =\frac{scale}{actual}=\frac{4}{l}\\ Width& =\frac{scale}{actual}=\frac{8}{w}\end{array}$$

  播放段落

  Then, write two proportions by setting the two length ratios equal to one another and the two width ratios equal to one another. The given unit is feet.
  ::然后写出两个比例, 设定两个长度比对等, 两个宽比对等。 给定单位是脚 。

  $$\begin{array}{rl}Length& =\frac{4}{l}=\frac{1}{160}\\ Width& =\frac{8}{w}=\frac{1}{160}\end{array}$$

  播放段落

  Next, cross multiply each equation.
  ::下一个,交叉乘以每个方程。

  $$\begin{array}{rl}L& =4\times 160\\ L& =640feet\\ W& =8\times 160\\ W& =1,280feet\end{array}$$

  播放段落

  Then, to find the area of the display, multiply length times width.
  ::然后,要找到显示区域,要找到显示区域,需要乘以长度乘以宽度。

  $$\begin{array}{r}640ft.\times 1,280ft.=819,200sq.ft.\end{array}$$

  播放段落

  The answer is 819,200 square feet.
  ::答案是819,200平方英尺。

  播放段落

  Example 2
  ::例2

  播放段落

  A small airplane has a wing span of 16 feet. The scale factor for a model is  $\begin{array}{r}\frac{1}{48}\end{array}$ . Find the wing span of the model in inches.  
  ::小型飞机的机翼宽度为16英尺。模型的大小因数是148。模型的机翼宽度以英寸计。

  播放段落

  First, write the scale factor.
  ::首先,写下比额表系数。

  $$\begin{array}{r}\frac{scale}{actual}=\frac{1}{48}\end{array}$$

  播放段落

  Next, write a ratio to represent the actual length of the wingspan to the unknown scale length of the wingspan.
  ::接下来,写一个比例,以表示翼的实际长度与翼的未知规模长度。

  $$\begin{array}{r}\frac{scale}{actual}=\frac{w}{16}\end{array}$$

  播放段落

  Then, write a proportion by setting the two ratios equal to one another. Remember that feet is the only unit given.  
  ::然后, 写一个比例, 设定两个比例对等 。 记住脚是给定的唯一单位 。

  $$\begin{array}{r}\frac{1}{48}=\frac{w}{16}\end{array}$$

   Next, cross multiply.

  $$\begin{array}{r}48w=16\end{array}$$

  $$\begin{array}{r}w=\frac{1}{3}\end{array}$$

  播放段落

  Then,  . 
  ::然后,然后,。 。 。

  $$\begin{array}{r}\frac{1}{3}foot\times \frac{12}{1}\frac{inches}{foot}=4inches\end{array}$$

  播放段落

  The answer is 4 inches. The wingspan on the model plane will be 4 inches long.
  ::答案是4英寸 机翼在模型平面上长4英寸

  播放段落

  Example 3
  ::例3

  播放段落

  The longest side of a triangular flower bed is 5.5   long. Leah wants to make a scale drawing of the flower bed. The scale factor for her drawing will be  $\begin{array}{r}\frac{1}{20}\end{array}$ . What will be the length of the longest side of the flower bed in her drawing?
  ::三角花床最长的一面是5.5长。 莉亚想要绘制花床的标尺。 她的标尺系数是 120。 绘制花床中最长的一面的长度是多少?

  播放段落

  First, write the scale factor.
  ::首先,写下比额表系数。

  $$\begin{array}{r}\frac{scale}{actual}=\frac{1}{20}\end{array}$$

  播放段落

  Next, write a ratio to represent the actual length of the side to the unknown scale length of the side.
  ::下一步,写一个比例,以表示侧的实际长度与侧的未知比例长度。

  $$\begin{array}{r}\frac{scale}{actual}=\frac{l}{5.5}\end{array}$$

  播放段落

  Then, write a proportion by setting the two ratios equal to one another. Meters is the only unit given. 
  ::然后,通过设定两个对等比率来写一个比例。 计算器是唯一给定的单位 。

  $$\begin{array}{r}\frac{1}{20}=\frac{l}{5.5}\end{array}$$

  meters

  播放段落

  Next, cross multiply.
  ::下一个,交叉乘数。

  $$\begin{array}{rl}5.5& =20l\\ l& =0.275meters\end{array}$$

  
  

  播放段落

  Then, convert meters to  .
  ::然后把仪表转换成

  $$\begin{array}{r}0.275meters\times \frac{100}{1}\frac{cm}{m}=2.75cm\end{array}$$

  播放段落

  The answer is 27.5 centimeters.
  ::答案是27.5厘米。

  播放段落

  Example 4
  ::例4

  播放段落

  An ant that Alison observed was too small to draw at its actual size. So, Alison made the scale drawing shown below. The scale factor for the drawing is 5. Find the actual length of the ant Alison observed.
  ::艾莉森观察到的蚂蚁太小,无法按其实际大小进行抽取。因此艾莉森绘制了下文所示的比重图。绘图的比重系数是5. 找出观察到的蚂蚁的实际长度。

  

  [Figure 2]

  播放段落

  First, write the scale factor.
  ::首先,写下比额表系数。

  $$\begin{array}{r}\frac{scale}{actual}=\frac{5}{1}\end{array}$$

  播放段落

  Next, write a ratio to represent the unknown actual length of the ant to the scale length of the ant.
  ::下一步,写一个比例,以表示未知的蚂蚁实际长度与蚂蚁规模长度。

  $$\begin{array}{r}\frac{scale}{actual}=\frac{2.75}{l}\end{array}$$

  播放段落

  Then, write a proportion by setting the two ratios equal to one another. Centimeters is the only unit given.
  ::然后,通过设定两个对等比率来写一个比例。 给出的单位只有厘米。

  $$\begin{array}{r}\frac{5}{1}=\frac{2.75}{1}cm\end{array}$$

  播放段落

  Next, cross multiply.
  ::下一个,交叉乘数。

  $$\begin{array}{rl}2.75& =5l\\ l& =0.55\end{array}$$

  $$\begin{array}{r}l=0.55cm\end{array}$$

  播放段落

  Then, convert centimeters to millimeters.
  ::然后把厘米转换成毫米

  $$\begin{array}{r}0.55cm\times \frac{10}{1}\frac{mm}{cm}=5.5mm\end{array}$$

  播放段落

  Example 5
  ::例5

  播放段落

  The scale drawing below shows a rectangular carpet. The scale factor for the drawing is  $\begin{array}{r}\frac{1}{24}\end{array}$ . What is the area, in square feet, of the actual carpet?
  ::下面的缩放比例显示长方形地毯。绘制比例系数是1 24。实际地毯的平方英尺面积是多少?

  

  [Figure 3]

  播放段落

  First, write the scale factor.
  ::首先,写下比额表系数。

  $$\begin{array}{r}\frac{scale}{actual}=\frac{1}{24}\end{array}$$

  播放段落

  Next, write ratios to represent the unknown actual length and width of the carpet to the scale length and width of the carpet.
  ::下一步,写出比例代表地毯实际长度和宽度与地毯大小长度和宽度之间未知的地毯实际长度和宽度。

  $$\begin{array}{r}\frac{scale}{actual}=\frac{4}{1}\frac{scale}{actual}=\frac{2}{w}\end{array}$$

  播放段落

  Then, write two proportions by setting the two length ratios equal to one another and the two width ratios equal to one another. The given unit is inches.
  ::然后写出两个比例, 设定两个长度比对等, 两个宽比对等。 给定单位是英寸 。

  $$\begin{array}{rl}Length& =\frac{4}{l}=\frac{1}{24}\\ Width& =\frac{2}{w}=\frac{1}{24}\end{array}$$

  播放段落

  Next, cross multiply each equation.
  ::下一个,交叉乘以每个方程。

  $$\begin{array}{rl}L& =4\times 24\\ L& =96inches\\ W& =2\times 24\\ W& =48inches\end{array}$$

  播放段落

  Then, to find the area of the carpet, multiply length x width.
  ::然后,为了找到地毯的区域, 乘以长度 x 宽度 。

  播放段落

    $\begin{array}{r}96"\times 48"=4,608sq.in.\end{array}$
  ::96 " ×48 " = 4, 608 s q. i n.

  播放段落

  The answer is 4,608 square inches.
  ::答案是4 608平方英寸

  播放段落

  Review
  ::回顾

  播放段落

  Solve each problem. Keep in mind that several problems may have more than one part.
  ::解决每个问题。 记住, 几个问题可能不止一个部分。

  播放段落
  1. Calvin drew a map of his neighborhood. The scale factor he used for his map was  $\begin{array}{r}\frac{1}{800}\end{array}$ . The actual distance between Calvin's house and his best friend Frank's house is 80 meters. What should be the distance, in centimeters, between those two places on his scale drawing?
     ::Calvin绘制了他的邻居的地图。他的地图所使用的比例系数是1800。Calvin的房子和他最好的朋友Frank的房子之间的实际距离是80米。这两个地方之间的距离,以厘米计,应该有多大?
  播放段落
  2. If the distance from Calvin’s house to the park is 40 meters, what would be the distance in centimeters?
     ::如果从Calvin家到公园的距离是40米,
  播放段落
  3. If the distance from Calvin’s house to the corner store is double the distance from his house to Frank’s, what would be the distance on the map in centimeters?
     ::如果从Calvin家到街角商店的距离是他家到Frank家的距离的两倍,那么地图上以厘米计的距离是多少?
  播放段落
  4. If the distance from Calvin’s house to his Grandmother’s was halfway between his and Frank’s, what would be the distance on the map in centimeters?
     ::如果从卡尔文家到他祖母家的距离是他和弗兰克家之间的一半,那么地图上以厘米计的距离是多少?
  播放段落
  5. Madeline built a model of a boat. The actual length of the boat is 24 feet. The scale factor she used for the model was  $\begin{array}{r}\frac{1}{36}\end{array}$ . What should be the length, in inches, of the model boat?
     ::Madeline建造了一条船的模型。船的实际长度是24英尺。她用于模型的尺度系数是136。模型船的长度(英寸)应该有多大?
  播放段落
  6. If the scale factor was  $\begin{array}{r}\frac{1}{72}\end{array}$ , what would the length be in inches of the model boat?
     ::如果比额表因数是172,那么模型船的长度是多少?
  播放段落
  7. A metal pipe is 2.5 meters long. Josh wants to make a scale drawing of the pipe. The scale factor for his drawing will be  $\begin{array}{r}\frac{1}{100}\end{array}$ . What will be the length, in centimeters, of the metal pipe in his drawing?
     ::金属管长2.5米。 Josh想对金属管做一个比例图。 其比例系数为 1100。 金属管的长度( 厘米) 是什么?
  播放段落
  8. Sydra observed a housefly that was too small to draw at its actual size. So she made a scale drawing, using 10 as a scale factor. The actual length of the housefly was 8 millimeters. What is the length of the housefly in Sydra's drawing in millimeters?
     ::Sydra观测到一只小到无法按实际大小抽取的软蝇。 所以她用10个比例系数做了一个比例图。 软蝇的实际长度是8毫米。 Sydra画的软蝇的长度是多少?
  播放段落
  9. What is the length of the housefly in Sydra's drawing in centimeters?
     ::Sydra画图时的 户外飞行长度是多少?
  播放段落
  10. Luis made a scale model of the doghouse he is going to build. The scale factor he used for this model was  $\begin{array}{r}\frac{1}{24}\end{array}$ . He wants the actual height of the doghouse to be 6 feet. What should be the height of the doghouse in his scale model?
      ::Luis做了一个他将要建造的狗舍的标尺模型。 他用于此模型的标尺系数是 12 24 。 他希望狗舍的实际高度是 6 英尺。 他的标尺模型中的狗舍的高度应该是 6 英尺 ?
  播放段落
  11. Jean-Marc used a scale factor of 5 to make this scale drawing of a moth.
      ::Jean-Marc用5的比值来绘制一个飞蛾的比值。

  

  [Figure 4]

  播放段落

  In the drawing, the wingspan of the moth measures 3 centimeters. What was the actual wingspan, in millimeters, of the moth Jean-Marc observed?
  ::在绘画中,飞蛾的翅膀面积相当于3厘米。 观察到的飞蛾Jean-Marc的实际翅膀面积(毫米)是多少?

  播放段落
  12. Below is a scale drawing of a swimming pool. In the scale drawing, the diameter of the pool measures  $\begin{array}{r}1\frac{1}{2}\end{array}$  inches. The scale factor for the drawing is  $\begin{array}{r}\frac{1}{72}\end{array}$ . What is the actual diameter of the pool in feet?
      ::下面是游泳池的一个比例图。在比例图中,游泳池的直径是112英寸。绘图的比重系数是172英寸。游泳池足部的实际直径是多少?

  

  [Figure 5]

  播放段落
  13. Barbara made a scale model of the Washington Monument. The scale factor for her model is  $\begin{array}{r}\frac{1}{1332}\end{array}$ . The height of her model is 5 inches. What is the actual height, in feet, of the Washington Monument?
      ::芭芭拉制作了华盛顿纪念碑的标尺模型。模型的标尺系数是1332。模型的高度是5英寸。华盛顿纪念碑的实际高度是多少?
  播放段落
  14. Below is a map of a city park. This map was created using a scale factor of  $\begin{array}{r}\frac{1}{300}\end{array}$ .
      ::下面是一座城市公园的地图。该地图是用比例因数 1 300 创建的。

  

  [Figure 6]

  播放段落
  1. On the map, the distance between the sandbox and the swings is 2.5 centimeters. What is the actual distance between the sandbox and the swings in meters?
     ::在地图上,沙箱和秋千之间的距离是2.5厘米。沙箱和秋千之间的实际距离是多少?
  播放段落
  2. On the map, the distance between the sandbox and the jungle gym is 1.7 centimeters. What is the actual distance between the sandbox and the jungle gym in meters?
     ::在地图上,沙箱和丛林健身房之间的距离是1.7厘米。 沙箱和丛林健身房之间的实际距离是多少?

  播放段落
  15. The scale drawing below shows the floor of Julian's bedroom.
      ::下面的尺度图显示了朱利安卧室的地板。

  

  [Figure 7]

  播放段落

  The scale factor for the drawing is  $\begin{array}{r}\frac{1}{200}\end{array}$ .
  ::加权系数为1 200。

  播放段落
  1. What are the dimensions, in meters, of the actual floor?
     ::实际地面的尺寸(以米计)是多少?
  播放段落
  2. What is the area, in square meters, of the actual floor?
     ::实际地面面积(平方米)是多少?

  播放段落

  Review (Answers)
  ::回顾(答复)

  播放段落

  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 。

  播放段落

  Resources
  ::资源