4.9 阅读和解释涉及距离和地区的地图

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  与距离和地区有关的阅读和口译地图

  

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  Kaitlyn is looking at a map and wants to find the area of the state of Nevada, United States. The map shows a scale of  $\begin{array}{r}{\frac{1}{2}}^{\mathrm{\prime }\mathrm{\prime }}\end{array}$  to 20 miles. If the distance on the map is $\begin{array}{r}{8}^{\mathrm{\prime }\mathrm{\prime }}\end{array}$  wide and $\begin{array}{r}{12.25}^{\mathrm{\prime }\mathrm{\prime }}\end{array}$  long, what is the area, in square miles of Nevada?
  ::Kaitlyn正在看一张地图,想要找到美国内华达州地区。地图显示的距离是12英里到20英里。如果地图上的距离是8英里宽和12.25英里长,那么内华达平方英里的面积是多少?

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  In this concept, you will learn to read and interpret maps involving distance and area by using scale measurement.
  ::在这个概念中,你们将学会使用比例尺测量法,读懂和解释与距离和面积有关的地图。

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  Distance and Area of Maps
  ::地图的距离和地图面积

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  A map is another type of scale drawing of a region. Maps can be very detailed or very simple, showing only points of interest and distances. You can read a map just like any other scale drawing—by using the scale.
  ::地图是区域比例图的另一种类型。地图可以非常详细或简单,只显示感兴趣的点和距离。您可以使用比例表阅读地图,就像任何其他比例图一样。

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  Let’s look at an example.
  ::让我们举个例子。

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  On the map below, the straight-line distance between San Francisco and San Diego is 3 inches . What is the actual straight-line distance between San Francisco and San Diego?
  ::在下面的地图上,旧金山和圣地亚哥之间的直线距离是3英寸。 旧金山和圣地亚哥之间的实际直线距离是多少?

  

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  First, set up a proportion. The scale in the drawing says that  $\begin{array}{r}0.5\text{inches}=75\text{miles}\end{array}$ , therefore the proportion is:
  ::首先,设定一个比例。绘图中的比例表示0.5英寸=75英里,因此比例是:

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  $$\begin{array}{r}\frac{0.5\text{inches}}{75\text{miles}}\end{array}$$

  
  ::0.5英寸75英里

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  Next, write the second ratio. You know the scale length is 3 inches. The unknown length is $\begin{array}{r}x\end{array}$ .
  ::接下来,写第二个比例。你知道缩放长度是3英寸。未知的长度是 x。

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  $$\begin{array}{r}\frac{0.5\text{inches}}{75\text{miles}}=\frac{3\text{inches}}{x\text{miles}}\end{array}$$

  
  Then, cross-multiply.
  ::0.5英寸75英里=3英寸x英里 然后 交叉倍增

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  $$\begin{array}{r}\begin{array}{rcl}\frac{0.5}{75}& =& \frac{3}{x}\\ 0.5x& =& 75\times 3\\ 0.5x& =& 225\end{array}\end{array}$$

  Then, divide both sides by 0.5 to solve for $\begin{array}{r}x\end{array}$ .
  ::0.575=3x0.5x=75x30.5x=225 然后,将两边除以0.5以解析 x。

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  $$\begin{array}{r}\begin{array}{rcl}0.5x& =& 225\\ \frac{0.5x}{0.5}& =& \frac{225}{0.5}\\ x& =& 450\end{array}\end{array}$$

  
  The answer is 450.
  ::0.5x=2250.5x0.5=2250.5x=450 答案是450。

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  The straight-line distance from San Francisco to San Diego is 450 miles.
  ::从旧金山到圣地亚哥的直线距离是450英里。

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  Note: The straight-line distance is also known as “as the crow flies.” If you were actually traveling from San Francisco to San Diego, it would be farther than 450 miles, since you would need to drive on highways that are not necessarily in a straight line.
  ::注意:直线距离也被称为“乌鸦飞翔 ” 。 如果你真的从旧金山前往圣地亚哥,那么距离会远超过450英里,因为你需要乘坐不一定是直线的高速公路。

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  You can also use a scale to find the area of a space or region. First, you need to figure out the length and width then we can complete any necessary calculations. Sometimes, you will have two different distances or areas that you are working to compare. When this happens, you can use proportions to compare the differences and similarities.
  ::您也可以使用一个比例来查找一个空间或区域的面积。首先,您需要找出长度和宽度,然后我们可以完成任何必要的计算。有时,您将有两个不同的距离或区域,或者您正在比较的区域。如果发生这种情况,您可以使用比例来比较差异和相似性。

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  Let’s look at an example.
  ::让我们举个例子。

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  Marta has a square with a side length of 4 inches. She has a similar square with dimensions that are twice the first square. How does the area of the larger square compare to the area of the smaller square?
  ::玛尔塔有一个方形,侧长4英寸。她的方形相似,尺寸是第一个方形的两倍。较大的方形面积与较小方形面积相比如何?

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  First, find the dimensions of the larger square. The problem states that the dimensions are twice the first square. You can use this information to figure out the scale factor, and this means they are scaled up by a factor of 2. The side length of the larger square is:
  ::首先,找到更大的方块的维度。 问题显示, 维度是第一个方块的两倍。 您可以使用此信息来解析比例系数, 这意味着它们会以2. 倍扩大。 较大方块的侧长是 :

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  $\begin{array}{r}4\text{inches}\times 2=8\text{inches}\end{array}$ .
  ::4英寸x2=8英寸。

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  Next, find the area of both squares and compare.
  ::接下来,找到两个广场的面积并进行比较。

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  $$\begin{array}{r}\begin{array}{rcl}& & \text{Area of smaller square}\text{Area of larger square}\\ & & \text{A = lw}\text{A = lw}\\ & & \text{A = (4 inches)(4 inches)}\text{A = (8 inches)(8 inches)}\\ & & \text{A}=16{\text{in}}^2\text{A}=64{\text{in}}^2\end{array}\end{array}$$

  
  ::面积小的平方面积面积小的面积A = 大方A = Lw A = LwA = (4英寸(4英寸)A = (8英寸)(8英寸)A = 16 英寸2 A= 64 英寸2

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  Then, compare the two areas. You want to know how the area of the larger square compares to the area of the smaller square. Write a ratio comparing the two areas. 
  ::然后比较这两个区域。 您想知道大方的面积与小方的面积相比如何。 写一个比较这两个区域的比例 。

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  $$\begin{array}{r}\frac{64{\text{in}}^2}{16{\text{in}}^2}=4\end{array}$$

  
  ::64216英寸2=4

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  The answer is 4.
  ::答案是4。

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  The area of the larger square is 4 times larger than the area of the smaller square.
  ::大方块面积比小方块面积大4倍。

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  This leads to a rule when comparing areas of similar figures. The ratio of areas of similar figures is the square of the scale factor.
  ::这导致在比较类似数字的区域时形成一项规则,类似数字区域的比率是比额表系数的平方。

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were given a problem about  Kaitlyn and her interest in Nevada.
  ::早些时候,你得到一个问题 关于凯特琳和她对内华达州的兴趣。

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  First, set up a proportion. The scale in the drawing says that  $\begin{array}{r}\frac{1}{2}\text{ inch}=20\text{miles}\end{array}$ , therefore the proportion is:
  ::首先,设定一个比例。绘图中的比例表示12英寸=20英里,因此比例是:

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  $$\begin{array}{r}\frac{0.5\text{inch}}{20\text{miles}}\end{array}$$

  
  ::0.5英寸20英里

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  Next, write the ratio representing the length. You know the scale length is 8 inches. The unknown length is $\begin{array}{r}x\end{array}$ .
  ::下一步,写代表长度的比率。您知道缩放长度是8英寸。未知的长度是 x。

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  $$\begin{array}{r}\frac{0.5\text{inch}}{20\text{miles}}=\frac{8\text{inches}}{x\text{miles}}\end{array}$$

  
  ::0.5英寸20英里=8英寸x英里

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  Then, cross-multiply to solve for $\begin{array}{r}x\end{array}$ .
  ::然后,交叉倍数解答 x。

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  $$\begin{array}{r}\begin{array}{rcl}\frac{0.5}{20}& =& \frac{8}{x}\\ 0.5x& =& 20\times 8\\ 0.5x& =& 160\end{array}\end{array}$$

  
  Then, divide both sides by 0.5 to solve for $\begin{array}{r}x\end{array}$ .
  ::0.520=8x0.5x=20x80.5x=160 然后,将两边除以0.5以解析 x。

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  $$\begin{array}{r}\begin{array}{rcl}0.5x& =& 160\\ \frac{0.5x}{0.5}& =& \frac{160}{0.5}\\ x& =& 320\end{array}\end{array}$$

  
  ::0.5x=1600.50x0.5=1600.5x=320

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  The answer is 320.
  ::答案是320

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  The width of Nevada is 320 miles.
  ::内华达州的宽度是320英里

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  Then, write the ratio representing the length. You know the scale length is 12.25 inches. The unknown length is $\begin{array}{r}x\end{array}$ .
  ::然后写代表长度的比。 你知道缩放长度是12.25英寸。 未知的长度是 x。

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  $$\begin{array}{r}\frac{0.5\text{inch}}{20\text{miles}}=\frac{12.25\text{inches}}{x\text{miles}}\end{array}$$

  
  ::0.5英寸20英里=12.25英寸x英里

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  Then, cross-multiply to solve for $\begin{array}{r}x\end{array}$ .
  ::然后,交叉倍数解答 x。

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  $$\begin{array}{r}\begin{array}{rcl}\frac{0.5}{20}& =& \frac{12.25}{x}\\ 0.5x& =& 20\times 12.25\\ 0.5x& =& 245\end{array}\end{array}$$

  
  Then, divide both sides by 0.5 to solve for $\begin{array}{r}x\end{array}$ .
  ::0.520=12.25x0.5x=20x12.250.5x=245 然后,将两边除以0.5以解答 x。

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  $$\begin{array}{r}\begin{array}{rcl}0.5x& =& 245\\ \frac{0.5x}{0.5}& =& \frac{245}{0.5}\\ x& =& 490\end{array}\end{array}$$

  
  ::0.5x=2450.5x0.5=2450.5x=490

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  The answer is 490.
  ::答案是490

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  The length of Nevada is 490 miles.
  ::内华达州长490英里

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  Finally, find the area of Nevada. 
  ::最后,找到内华达州

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  $$\begin{array}{r}\begin{array}{rcl}& & \text{A = lw}\\ & & A=(320\text{miles})(490\text{miles})\\ & & A=156,800\text{square miles}\end{array}\end{array}$$

  
  ::A=lwA=(320英里(490英里)A=156 800平方英里)A=156 800平方英里

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  The answer is 156,800.
  ::答案是156 800

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  The area of Nevada is $\begin{array}{r}156,800{\text{mi}}^2\end{array}$ .
  ::内华达州面积为156 800米2。

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  Example 2
  ::例2

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  If the scale is $\begin{array}{r}0.5\text{inches}=100\text{miles}\end{array}$ , how many inches is 500 miles?
  ::如果比例尺是0.5英寸=100英里,那么500英里是多少英寸?

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  First, set up the proportion to solve.
  ::首先,设定解决比例。

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  $$\begin{array}{r}\frac{0.5\text{inches}}{100\text{miles}}=\frac{x\text{inches}}{500\text{miles}}\end{array}$$

  
  ::0.5 英寸100英里=x 英寸500英里

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  Next, cross multiply.
  ::下一个,交叉乘数。

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  $$\begin{array}{r}\begin{array}{rcl}\frac{0.5}{100}& =& \frac{x}{500}\\ 100x& =& 0.5\times 500\\ 100x& =& 250\end{array}\end{array}$$

  Then, divide by 100 to solve for $\begin{array}{r}x\end{array}$ .
  ::0.5100=x500100x=0.5x500100x=250,再除以100以解答 x。

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  $$\begin{array}{r}\begin{array}{rcl}100x& =& 250\\ \frac{100x}{100}& =& \frac{250}{100}\\ x& =& 2.5\end{array}\end{array}$$

  The answer is 2.5.
  ::100x=250100x100=250100x=250100x=2.5 答案是2.5。

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  The distance on the map will be 2.5 inches.
  ::地图上的距离是2.5英寸

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  Example 3
  ::例3

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  If $\begin{array}{r}{1}^{\mathrm{\prime }\mathrm{\prime }}=2000\text{miles}\end{array}$   , find each actual distance with a scale measurement of $\begin{array}{r}{3}^{\mathrm{\prime }\mathrm{\prime }}\end{array}$ .
  ::如果12000英里, 找到每个实际距离, 比例尺测量为 3。

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  First, set up a proportion. The scale in the drawing says that  $\begin{array}{r}1\text{inch}=2000\text{miles}\end{array}$ , therefore the proportion is:
  ::首先,设定一个比例。绘图中的比例表示 1 英寸= 2000 英里, 因此比例是 :

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  $$\begin{array}{r}\frac{1\text{inch}}{2000\text{miles}}\end{array}$$

  
  ::1英寸2000英里

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  Next, write the second ratio. You know the scale length is 3 inches. The unknown length is $\begin{array}{r}x\end{array}$ .
  ::接下来,写第二个比例。你知道缩放长度是3英寸。未知的长度是 x。

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  $$\begin{array}{r}\frac{1\text{inch}}{2000\text{miles}}=\frac{3\text{inches}}{x\text{miles}}\end{array}$$

  
  Then, cross-multiply to solve for $\begin{array}{r}x\end{array}$ .
  ::1 英寸2000 英里= 3 英寸x 英里 然后 交叉倍增 解决 x 。

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  $$\begin{array}{r}\begin{array}{rcl}\frac{1}{2000}& =& \frac{3}{x}\\ 1x& =& 2000\times 3\\ x& =& 6000\end{array}\end{array}$$

  
  ::12000=3x1x=2000x3x=6000

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  The answer is 6000.
  ::答案是6000

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  The actual distance is 6000 miles.
  ::实际距离是6000英里

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  Example 4
  ::例4

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  If $\begin{array}{r}{1}^{\mathrm{\prime }\mathrm{\prime }}=2000\text{miles}\end{array}$ , find each actual distance with a scale measurement of  $\begin{array}{r}{\frac{1}{2}}^{\mathrm{\prime }\mathrm{\prime }}\end{array}$ .
  ::如果12000英里, 找到每个实际距离, 比例尺测量为 12。

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  First, set up a proportion. The scale in the drawing says that  $\begin{array}{r}1\text{inch}=2000\text{miles}\end{array}$ , therefore the proportion is:
  ::首先,设定一个比例。绘图中的比例表示 1 英寸= 2000 英里, 因此比例是 :

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  $$\begin{array}{r}\frac{1\text{inch}}{2000\text{miles}}\end{array}$$

  
  ::1英寸2000英里

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  Next, write the second ratio. You know the scale length is $\begin{array}{r}{0.5}^{\mathrm{\prime }\mathrm{\prime }}\end{array}$  inches. The unknown length is $\begin{array}{r}x\end{array}$ .
  ::下一步写第二个比例。 您知道缩放长度是 0. 5 英寸。 未知的长度是 x 。

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  $$\begin{array}{r}\frac{1\text{inch}}{2000\text{miles}}=\frac{0.5\text{inches}}{x\text{miles}}\end{array}$$

  
  ::1英寸2000英里=0.5英寸x英里

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  Then, cross-multiply to solve for  $\begin{array}{r}x\end{array}$ .
  ::然后,交叉倍数解答 x。

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  $$\begin{array}{r}\begin{array}{rcl}\frac{1}{2000}& =& \frac{0.5}{x}\\ 1x& =& 2000\times 0.5\\ x& =& 1000\end{array}\end{array}$$

  The answer is 1000.
  ::12000=0.5x1x=2000x0.5x=1000 答案是1000。

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  The actual distance is 1000 miles.
  ::实际距离是1000英里

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  Example 5
  ::例5

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  If $\begin{array}{r}{1}^{\mathrm{\prime }\mathrm{\prime }}=2000\text{miles}\end{array}$ , find each actual distance with a scale measurement of  $\begin{array}{r}{\frac{1}{4}}^{\mathrm{\prime }\mathrm{\prime }}\end{array}$ .
  ::如果12000英里, 找到每个实际距离, 比例尺测量为 14。

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  First, set up a proportion. The scale in the drawing says that  $\begin{array}{r}1\text{ inch}=2000\text{ miles}\end{array}$ , therefore the proportion is:
  ::首先,设定一个比例。绘图中的比例表示 1 英寸= 2000 英里, 因此比例是 :

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  $$\begin{array}{r}\frac{1\text{inch}}{2000\text{miles}}\end{array}$$

  
  ::1英寸2000英里

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  Next, write the second ratio. You know the scale length is $\begin{array}{r}{0.25}^{\mathrm{\prime }\mathrm{\prime }}\end{array}$  inches. The unknown length is $\begin{array}{r}x\end{array}$ .
  ::接下来,写第二个比例。你知道缩放长度为0.25英寸。未知的长度是 x。

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  $$\begin{array}{r}\frac{1\text{inch}}{2000\text{miles}}=\frac{0.25\text{inches}}{x\text{miles}}\end{array}$$

  
  ::1英寸2000英里=0.25英寸x英里

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  Then, cross-multiply to solve for $\begin{array}{r}x\end{array}$ .
  ::然后,交叉倍数解答 x。

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  $$\begin{array}{r}\begin{array}{rcl}\frac{1}{2000}& =& \frac{0.25}{x}\\ 1x& =& 2000\times 0.25\\ x& =& 500\end{array}\end{array}$$

  The answer is 500.
  ::12000=0.25x1x=2000x0.25x=500 答案是500。

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  The actual distance is 500 miles.
  ::实际距离是500英里

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  Review
  ::回顾

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  Using the scale $\begin{array}{r}1\text{inch}=5.5\text{miles}\end{array}$ , figure out the number of inches needed to map each number of miles. Use proportions to figure out your answers.
  ::使用比例尺 1 英寸= 5. 5 英里, 算出绘制每一英里数所需的英寸数。 使用比例来计算答案 。

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  1. 16.5 miles
  ::1.5英里 16.5英里

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  2. 11 miles
  ::2. 11英里

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  3. 27.5 miles
  ::3. 27.5英里

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  4. 8.25 miles
  ::4. 8.25英里

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  5. 33 miles
  ::5. 33英里

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  6. 60.5 miles
  ::6. 60.5英里

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  7. 13.75 miles
  ::7.75英里 13.75英里

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  Using the scale $\begin{array}{r}0.5\text{inches}=100\text{miles}\end{array}$ , figure out the number of actual miles represented by each scale measurement.
  ::使用0.5英寸=100英里的尺度,算出每个尺度测量所代表的实际英里数。

  8.  $\begin{array}{r}{1}^{\mathrm{\prime }\mathrm{\prime }}\end{array}$

  9.  $\begin{array}{r}{2}^{\mathrm{\prime }\mathrm{\prime }}\end{array}$

  10.  $\begin{array}{r}{3}^{\mathrm{\prime }\mathrm{\prime }}\end{array}$

  11. $\begin{array}{r}{\frac{1}{4}}^{\mathrm{\prime }\mathrm{\prime }}\end{array}$

  12.   $\begin{array}{r}{\frac{3}{4}}^{\mathrm{\prime }\mathrm{\prime }}\end{array}$

  13.  $\begin{array}{r}1{\frac{1}{2}}^{\mathrm{\prime }\mathrm{\prime }}\end{array}$

  14. $\begin{array}{r}2{\frac{1}{2}}^{\mathrm{\prime }\mathrm{\prime }}\end{array}$

  15.  $\begin{array}{r}5{\frac{1}{2}}^{\mathrm{\prime }\mathrm{\prime }}\end{array}$

  16.  $\begin{array}{r}{7}^{\mathrm{\prime }\mathrm{\prime }}\end{array}$

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  Review (Answers)
  ::回顾(答复)

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  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 。

   

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  Resources
  ::资源