Course: 4.8 阅读和口译比例绘图和下限计划

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阅读和口译比例绘图和下限计划
The local community is planning to put a playground on a block of unused land. This is the drawing that they have created. You can see that the width of the playground is 7 inches wide in the drawing. If this is the case, what is the actual width of the playground?
::当地社区正计划在一块未使用的土地上安放一个游乐场。这就是他们创造的图画。您可以看到游戏场的宽度在绘画中是7英寸宽。如果是这样, 那么操场的实际宽度是多少 ?

In this concept, you will learn to interpret scale drawings and floor plans.
::在这个概念中,你将学会解释比额表图和平面图。

Scale Drawing
::缩平绘图

A scale drawing or floor plan is a representation of an actual object or space drawn in two-dimensions. For a floor plan, you can imagine that you are directly above the building looking down. The lines represent the walls of the building, and the space in between the lines represents the floor.
::平面图或平面图代表一个实际的物体或用两层图画的空间。对于平面图,你可以想象你直接在楼顶向下看。平面图代表建筑物的墙壁,两层线之间的空间代表地板。

In order to find the actual dimensions from a floor plan, you can set up and solve a proportion. The scale given in the drawing is the first ratio. The unknown length and the scale length is the second ratio.
::为了从平面图中找到实际尺寸,您可以设置和解决比例问题。绘图中给出的比重为第一比。长度未知,比重长度为第二比。

Let’s look at an example.
::让我们举个例子。

The floor plan below shows several classrooms at Craig’s school. The length of Classroom 2 in the floor plan is 2 inches. What is the actual length, in feet, of Classroom 2?
::下面的平面图显示克雷格学校的几间教室。平面图中第2教室的长度是2英寸。第二教室的实际足长是多少?

First, set up a proportion. The scale in the drawing says that $\begin{align*}{\frac{1}{2}}^{{\prime}{\prime}} = 3^{\prime}\end{align*}$ , therefore the proportion is:
::首先,设定一个比例。在绘图中,比例表上写着 1233,因此比例是:

$\begin{aligned} \frac{0.5 inches}{3 feet} \end{aligned}$
$\begin{align*}\frac{0.5 \text{ inches}}{3 \text{ feet}}\end{align*}$
::0.5英寸3英尺

Next, write the second ratio. You know the scale length is 2 inches. The unknown length is $\begin{align*}x\end{align*}$ . Make sure that the second ratio follows the form of the first ratio: inches over feet.
::下一步写第二个比例。您知道缩放长度是 2 英寸。未知的长度是 x。请确定第二个比例遵循第一个比例的形式: 跨脚英寸。

$\begin{aligned} \frac{0.5 inches}{3 feet} = \frac{2 inches}{x feet} \end{aligned}$
$\begin{align*}\frac{0.5 \text{ inches}}{3 \text{ feet}}=\frac{2 \text{ inches}}{x \text{ feet}}\end{align*}$ Then, cross-multiply.
::0.5英寸3英尺=2英寸x英尺然后交叉倍增

$\begin{aligned} \begin{array}{rcl} \frac{0.5}{3} & = & \frac{2}{x} \\ 0.5 x & = & 2 \times 3 \\ 0.5 x & = & 6 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \frac{0.5}{3} &=& \frac{2}{x} \\ 0.5 x &=& 2 \times 3 \\ 0.5 x &=& 6 \end{array}\end{align*}$ Then, divide both sides by 0.5 to solve for $\begin{align*}x\end{align*}$ .
::0.53=2x0.5x=2x30.5x=6 然后,将两边除以0.5以解析 x。

$\begin{aligned} \begin{array}{rcl} 0.5 x & = & 6 \\ \frac{0.5 x}{0.5} & = & \frac{6}{0.5} \\ x & = & 12 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 0.5 x &=& 6\\ \frac{0.5 x}{0.5} &=& \frac{6}{0.5} \\ x &=& 12 \end{array}\end{align*}$ The answer is 12.
::0.5x=60.5x0.5=60.5x=12 答案是12。

The actual length of the classroom is 12 feet.
::教室的实际长度为12英尺。

Examples
::实例

Example 1
::例1

Remember the playground to scale?
::记得游乐场吗?

Consider the earlier drawing of the playground’s scale: $\begin{align*}1^{{\prime}{\prime}}= 20^{\prime}\end{align*}$ .
::考虑更早地绘制游乐场规模: 120。

First, set up a proportion. The scale in the drawing says that $\begin{align*}1^{{\prime}{\prime}}= 20^{\prime}\end{align*}$ , therefore the proportion is:
::首先,设定一个比例。绘图中的比重表示 1 20 , 因此比例是 :

$\begin{aligned} \frac{1 inch}{20 feet} \end{aligned}$
$\begin{align*}\frac{1 \text{ inch}}{20 \text{ feet}}\end{align*}$
::1英寸20英尺

Next, write the second ratio. You know the scale length is $\begin{align*}7^{{\prime}{\prime}}\end{align*}$ inches. The unknown length is $\begin{align*}x\end{align*}$ .
::下一步写第二个比例。您知道缩放长度是 7 英寸。未知的长度是 x 。

$\begin{aligned} \frac{1 inch}{20 feet} = \frac{7 inches}{x feet} \end{aligned}$
$\begin{align*}\frac{1 \text{ inch}}{20 \text{ feet}}=\frac{7 \text{ inches}}{x \text{ feet}}\end{align*}$ Then, cross-multiply to solve for $\begin{align*}x\end{align*}$ .
::1 英寸20英尺= 7 英寸x 英尺,然后, 交叉倍数解答 x 。

$\begin{aligned} \begin{array}{rcl} \frac{1}{20} & = & \frac{7}{x} \\ 1 x & = & 20 \times 7 \\ x & = & 140 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \frac{1}{20}&=&\frac{7}{x} \\ 1x &=& 20 \times 7 \\ x &=& 140 \end{array}\end{align*}$
The answer is 140.
::120=7x1x=20x7x=140 答案是 140。

The actual width of the playground is 140 feet.
::游乐场的实际宽度为140英尺。

Example 2
::例2

The image below shows the fountain in front of a hotel. The diameter of the fountain is 4 . What is the actual diameter of the fountain?
::喷泉直径是4。喷泉的实际直径是多少?

First, set up a proportion. The scale for the image says that $\begin{align*}1 \ cm = 0.5 \ m\end{align*}$ , therefore the proportion is:
::首先,设定比例。图像的缩放比例表示 1厘米=0.5米,因此比例是:

$\begin{aligned} \frac{1 c m}{0.5 m} \end{aligned}$
$\begin{align*}\frac{1 \ cm}{0.5 \ m}\end{align*}$
::1厘米0.5米

Next, write the second ratio. You know the scale length is 4 cm . The unknown length is $\begin{align*}x\end{align*}$ .
::下一步写第二个比例。你知道缩放长度是 4 厘米。未知的长度是 x 。

$\begin{aligned} \frac{1 c m}{0.5 m} = \frac{4 c m}{x m} \end{aligned}$
$\begin{align*}\frac{1 \ cm}{0.5 \ m}=\frac{4 \ cm}{x \ m}\end{align*}$ Then, cross-multiply to solve for $\begin{align*}x\end{align*}$ .
::1cm0.5 m=4 cmx m 然后, 交叉倍数解答 x。

$\begin{aligned} \begin{array}{rcl} \frac{1}{0.5} & = & \frac{4}{x} \\ 1 x & = & 0.5 \times 4 \\ x & = & 2 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{ rcl} \frac{1}{0.5} &=& \frac{4}{x} \\ 1x &=& 0.5\times4\\ x &=& 2 \end{array}\end{align*}$ The answer is 2.
::10.5=4x1x=0.5x4x=2 答案是2。

The actual diameter of the fountain is 2 m.
::喷泉的实际直径为2米。

Example 3
::例3

If the scale said $\begin{align*}\frac{1}{2^{\prime \prime}}= 4 \text{ feet}\end{align*}$ , what would be the inches drawn for a room $\begin{align*}8^\prime \times 12^\prime\end{align*}$ ?
::如果比例尺是12尺4英尺,那么8寸12寸房间的英寸是多少?

First, set up a proportion. The scale in the drawing says that $\begin{align*}\frac{1}{2}^{{\prime}{\prime}}= 4^{\prime}\end{align*}$ , therefore the proportion is:
::首先,设定一个比例。在绘图中,比例表上写着 1244,因此比例是:

$\begin{aligned} \frac{0.5 inches}{4 feet} \end{aligned}$
$\begin{align*}\frac{0.5 \text{ inches}}{4 \text{ feet}}\end{align*}$
::0.5英寸4英尺

Next, write the second ratio. You know the scale length is $\begin{align*}8^\prime \text{ inches}\end{align*}$ . The unknown length is $\begin{align*}x\end{align*}$ .
::下一步写第二个比例。您知道缩放长度是 8 英寸。未知的长度是 x 。

$\begin{aligned} \frac{0.5 inches}{4 feet} = \frac{x inches}{8 feet} \end{aligned}$
$\begin{align*}\frac{0.5 \text{ inches}}{4 \text{ feet}}=\frac{x \text{ inches}}{8 \text{ feet}}\end{align*}$
::0.5英寸4英尺=x英寸8英尺

Then use the same ratio to find the other unknown length.
::然后使用同样的比例来找到其他未知长度。

$\begin{aligned} \frac{0.5 inches}{4 feet} = \frac{x inches}{12 feet} \end{aligned}$
$\begin{align*}\frac{0.5 \text{ inches}}{4 \text{ feet}} = \frac{x \text{ inches}}{12 \text{ feet}}\end{align*}$
::0.5英寸4英尺=x 12英寸12英尺

Then, cross-multiply for both ratios.
::然后,交叉乘以两个比率。

$\begin{aligned} \begin{array}{rcl} \frac{0.5}{4} & = & \frac{x}{8} \frac{0.5}{4} = \frac{x}{12} \\ 4 x & = & 0.5 \times 8 4 x = 0.5 \times 12 \\ 4 x & = & 4 4 x = 6 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \frac{0.5}{4} & = & \frac{x}{8} \qquad \quad \frac{0.5}{4}=\frac{x}{12} \\ 4x & = & 0.5 \times 8 \quad \quad 4x = 0.5 \times 12 \\ 4x & = & 4 \qquad \qquad \ \ 4x = 6 \end{array}\end{align*}$
::0.54=80.54=x124x=0.5×84x=0.5×124x=4 4x=6

Then, divide both sides to solve for $\begin{align*}x\end{align*}$ .
::然后把两边分开解决x

$\begin{aligned} \begin{array}{rcl} 4 x & = & 4 4 x = 6 \\ \frac{4 x}{4} & = & \frac{4}{4} \frac{4 x}{4} = \frac{6}{4} \\ x & = & 1 x = 1.5 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 4x&=&4 \quad \quad \quad 4x = 6\\ \frac{4x}{4} &= &\frac{4}{4} \qquad \frac{4x}{4} =\frac{6}{4} \\ x &=& \ 1 \quad \qquad \ x = 1.5 \end{array}\end{align*}$
::4x=44x=64x4=444x4=64x=1x=1.5

The answers are 1 and 1.5.
::答案是1和1.5。

The scale dimensions are $\begin{align*}1 ^{{\prime}{\prime}} \times 1.5^{{\prime}{\prime}}\end{align*}$ .
::尺度尺寸为11.5。

Example 4
::例4

If the scale said $\begin{align*}\frac{1}{2^{{\prime}{\prime}}}=4 \text{ feet}\end{align*}$ , what would be the inches drawn for a room $\begin{align*}20^{\prime} \times 28^{\prime}\end{align*}$ ?
::如果比例尺是12尺4英尺, 20寸28寸房间的英寸是多少?

First, set up a proportion. The scale in the drawing says that $\begin{align*}{\frac{1}{2}}^{{\prime}{\prime}} = 4^{\prime}\end{align*}$ , therefore the proportion is:
::首先,设定一个比例。在绘图中,比例表上写着 1244,因此比例是:

$\begin{aligned} \frac{0.5 inches}{4 feet} \end{aligned}$
$\begin{align*}\frac{0.5 \text{ inches}}{4 \text{ feet}}\end{align*}$
::0.5英寸4英尺

Next, write the second ratio. You know the scale length is $\begin{align*}8^{\prime}\end{align*}$ inches. The unknown length is $\begin{align*}x\end{align*}$ .
::下一步写第二个比例。您知道缩放长度是 8 英寸。未知的长度是 x 。

$\begin{aligned} \frac{0.5 inches}{4 feet} = \frac{x inches}{20 feet} \end{aligned}$
$\begin{align*}\frac{0.5 \text{ inches}}{4 \text{ feet}}=\frac{x \text{ inches}}{20 \text{ feet}}\end{align*}$
::0.5英寸4英尺=x 20英寸20英尺

Then use the same ratio to find the other unknown length.
::然后使用同样的比例来找到其他未知长度。

$\begin{aligned} \frac{0.5 inches}{4 feet} = \frac{x inches}{28 feet} \end{aligned}$
$\begin{align*}\frac{0.5 \text{ inches}}{4 \text{ feet}}=\frac{x \text{ inches}}{28 \text{ feet}}\end{align*}$
::0.5英寸4英尺=x 28英寸28英尺

Then, cross-multiply for both ratios.
::然后,交叉乘以两个比率。

$\begin{aligned} \begin{array}{rcl} \frac{0.5}{4} & = & \frac{x}{20} \frac{0.5}{4} = \frac{x}{28} \\ 4 x & = & 0.5 \times 20 4 x = 0.5 \times 28 \\ 4 x & = & 10 4 x = 14 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \frac{0.5}{4}&=& \frac{x}{20} \qquad \qquad \frac{0.5}{4}=\frac{x}{28} \\ 4x &= & 0.5 \times 20 \qquad \ \ \ 4x = 0.5 \times 28 \\ 4x &=& 10 \qquad \quad \qquad 4x =14 \end{array}\end{align*}$
::0.54=x200.54=x284x=0.5x20 4x=0.5x284x=104x=14

Then, divide both sides to solve for $\begin{align*}x\end{align*}$ .
::然后把两边分开解决x

$\begin{aligned} \begin{array}{rcl} 4 x & = & 10 4 x = 14 \\ \frac{4 x}{4} & = & \frac{10}{4} \frac{4 x}{4} = \frac{14}{4} \\ x & = & 2.5 x = 3.5 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 4x&=&10 \qquad \quad \quad 4x= 14 \\ \frac{4x}{4} &= & \frac{10}{4} \qquad \quad \frac{4x}{4} = \frac{14}{4} \\ x &=& 2.5 \qquad \quad \ \ \ \ x = 3.5 \end{array}\end{align*}$
::4x=104x=144x4=104x4=104x4=104x4=144x4=144x=2.5x=3.5

The answers are 2.5 and 3.5. Therefore the scale dimensions are $\begin{align*}2.5^{{\prime}{\prime}} \times 3.5^{{\prime}{\prime}}\end{align*}$ .
::答案是2.5和3.5,因此,尺度尺寸是2.5和3.5。

Example 5
::例5

If the scale said $\begin{align*}\frac{1}{ 2^{\prime \prime}}= 4 \text{ feet}\end{align*}$ , what would be the inches drawn for a room $\begin{align*}16^\prime \times 24^\prime\end{align*}$ ?
::如果比例尺是12尺4英尺, 16寸24寸房间的英寸是多少?

First, set up a proportion. The scale in the drawing says that $\begin{align*}\frac{1}{ 2}^{\prime \prime}= 4^\prime\end{align*}$ , therefore the proportion is:
::首先,设定一个比例。在绘图中,比例表上写着 1244,因此比例是:

$\begin{aligned} \frac{0.5 inches}{4 feet} \end{aligned}$
$\begin{align*}\frac{0.5 \text{ inches}}{4 \text{ feet}}\end{align*}$
::0.5英寸4英尺

Next, write the second ratio. You know the scale length is $\begin{align*}8^\prime \text{ inches}\end{align*}$ . The unknown length is $\begin{align*}x\end{align*}$ .
::下一步写第二个比例。您知道缩放长度是 8 英寸。未知的长度是 x 。

$\begin{aligned} \frac{0.5 inches}{4 feet} = \frac{x inches}{16 feet} \end{aligned}$
$\begin{align*}\frac{0.5 \text{ inches}}{4 \text{ feet}}=\frac{x \text{ inches}}{16 \text{ feet}}\end{align*}$
::0.5英寸4英尺=x 16英寸16英尺

Then use the same ratio to find the other unknown length.
::然后使用同样的比例来找到其他未知长度。

$\begin{aligned} \frac{0.5 inches}{4 feet} = \frac{x inches}{24 feet} \end{aligned}$
$\begin{align*}\frac{0.5 \text{ inches}}{4 \text{ feet}}=\frac{x \text{ inches}}{24 \text{ feet}}\end{align*}$ Then, cross-multiply for both ratios.
::0.5 英寸4英尺=x 英寸24, 然后, 交叉乘以两个比例。

$\begin{aligned} \begin{array}{rcl} \frac{0.5}{4} & = & \frac{x}{16} \frac{0.5}{4} = \frac{x}{24} \\ 4 x & = & 0.5 \times 16 4 x = 0.5 \times 24 \\ 4 x & = & 8 4 x = 12 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \frac{0.5}{4}&=&\frac{x}{16} \qquad \quad \quad \frac{0.5}{4}=\frac{x}{24} \\ 4x &= &0.5 \times 16 \qquad \ \ \ 4x = 0.5 \times 24 \\ 4x &=& 8 \qquad \qquad \quad \ 4x =12 \end{array}\end{align*}$
::0.54=x160.54=x244x=0.5x16 4x=0.5x244x=8 4x=12

Then, divide both sides to solve for $\begin{align*}x\end{align*}$ .

$\begin{aligned} \begin{array}{rcl} 4 x & = & 8 4 x = 12 \\ \frac{4 x}{4} & = & \frac{8}{4} \frac{4 x}{4} = \frac{12}{4} \\ x & = & 2 x = 3 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 4x&=&8 \quad \quad \quad 4x =12 \\ \frac{4x}{4} &= & \frac{8}{4} \qquad \frac{4x}{4} = \frac{12}{4} \\ x &=& 2 \qquad \quad \ \ x = 3 \end{array}\end{align*}$
The answers are 2 and 3.
::然后,将两边分隔为 x. 4x=84x=124x4=844x4=124x2x=3, 答案是 2和3。

The scale dimensions are $\begin{align*}2^{{\prime}{\prime}} \times 3^{{\prime}{\prime}}\end{align*}$ .
::尺度尺寸为 233。

Review
::回顾

This floor plan shows Bonnie’s house. Use it to answer the following questions.
::这个楼层图显示Bonnie的房子, 用它回答下列问题。

1. The width of the bedroom on the floor plan measures 1.5 inches. What is the actual width of the bedroom?
::1. 地板平面图上卧室的宽度为1.5英寸,卧室的实际宽度是多少?

2. The length of the kitchen on the floor plan measures 3 inches. What is the actual length of the kitchen?
::2. 平面平面平面平面平面平面平面平面平面平面平面的厨房长度为3英寸,实际厨房长度是多少?

3. The study measures 2 inches by 1.5 inches on the floor plan. What is the actual area of the study?
::3. 研究用平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面。

4. The study measures 2 inches by 1.5 inches on the floor plan. What is the actual perimeter of the study?
::4. 研究用平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面。

5. The study measures 2 inches by 1.5 inches on the floor plan. What are the actual dimensions of the study?
::5. 研究用平面图衡量2英寸乘1.5英寸,研究的实际层面是什么?

6. If the length of the bedroom is the same as the study, what are the actual dimensions of the bedroom?
::6. 如果卧室的长度与研究相同,卧室的实际尺寸是多少?

7. What is the area of the bedroom?
::7. 卧室的面积是多少?

8. What is the actual length of the outside of the house?
::8. 房外的实际长度是多少?

9. A square measures 10 inches on each side. What is its area?
::9. 平方等于每一方10英寸,面积是多少?

10. If the scale is $\begin{align*}1^ {{\prime}{\prime}}= 50 \ ft\end{align*}$ , what is the actual side length of the same square?
::10. 如果比额表为150英尺,则同一平方的实际边长是多少?

11. What is its area?
::11. 它的面积是什么?

12. A classroom measures 21 feet by 15 feet. A second classroom has a similar shape but the dimensions are $\begin{align*}\frac{1}{5}\end{align*}$ as long. How does the area of the second classroom compare to the area of the first classroom?
::12. 教室面积为21英尺乘15英尺,第二个教室形状相似,但尺寸为15英尺,第二个教室面积与第一个教室面积相比如何?

13. What are the dimensions of the second classroom?
::13. 第二个教室的层面是什么?

14. Yuri’s backyard has an area of 1,000 square feet. The dimensions of Kyle’s backyard are all $\begin{align*}\frac{1}{5}\end{align*}$ the size of Yuri’s. What is the area of Kyle’s backyard?
::14. 尤里后院的面积为1,000平方英尺,凯尔后院的面积都是尤里的15尺。凯尔后院的面积是多少? 凯尔后院的面积是多少?

15. Mary’s yard is double the area of Kyle’s backyard. What is the area of Mary’s yard?
::15. 玛丽的院子是凯尔后院面积的两倍,玛丽的院子面积是多少?

what would be the inches drawn for a room

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

阅读和口译比例绘图和下限计划

Scale Drawing ::缩平绘图

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 Drawing
::缩平绘图

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