节: 解决问题时使用单位缩放 | 4.6 解决问题时使用单位规模

章节大纲

Josh is building a scale diagram of Mount Everest. He knows that Mount Everest is 29,035 feet high. He doesn’t want his model to be too large so thinks he should use $\begin{aligned} \frac{1 i n}{2000 f t} \end{aligned}$ . If Josh uses this scale, will his model be too big?
::Josh正在绘制珠穆朗玛峰比例图。他知道珠穆朗玛峰高达29,035英尺。他不希望他的模型太大,因此认为他在2000年应该使用1英尺。如果Josh使用这个比例图,那么他的模型会不会太大?

In this concept, you will learn to use unit scale when problem solving.
::在此概念中,您将学会在解决问题时使用单位规模。

Unit Scale
::单位比例缩缩

Sometimes in life, you have a real-life object that you want to represent in a smaller form. Think about buildings. When an architect is planning a building design, he or she may design a model of the building. This model shows the dimensions of the building in a smaller way. When you do this, you take the actual dimensions and shrink them down to build a model. When you do this, you create a unit scale for the model. When you create a unit scale, you decide on an appropriate measurement to represent an actual measurement.
::有时在生活中,你有一个真实的物体,你想以较小的形式代表它。想一想建筑物。当一个建筑设计师正在规划建筑设计时,他(她)可以设计一个建筑模型。这个模型以较小的方式显示建筑物的尺寸。做这个模型时,你采取实际的尺寸,将其缩放,以构建一个模型。做这个模型时,你为模型创建一个单位尺度。当创建一个单位尺度时,你将决定一个适当的计量,以代表实际的测量。

First, let’s look at a unit scale.
::首先,让我们看看单位规模。

$\begin{aligned} 1 inch = 3 feet \end{aligned}$
::1英寸=3英尺

This is a unit scale. You have a unit represented by the one inch. Remember that when you talk about unit, you are talking about a relationship to one. You have one inch represented by three feet.
::这是单位比例表。您有一个单位, 代表单位为一英寸。记住, 当您谈论单位时, 您正在谈论与一个单位的关系。您的单位为一英寸, 代表三英尺。

The one inch is the scale dimension and the three feet is the actual dimension we are measuring.
::一英寸是尺度尺寸,三英尺是实际的尺寸,我们正在测量。

Now, not all objects that you will create a model of will measure exactly what the unit scale does, so you have to use a unit scale to show the relationship between scale dimensions and actual dimensions . Scale dimensions are the dimensions of the model, and actual dimensions are the real–life dimensions.
::现在,不是所有您要创建的模型都能够精确测量单位规模的精确度量,所以您必须使用单位规模来显示尺度大小与实际尺寸之间的关系。规模大小是模型的维度,而实际维度是实际生命维度。

Let’s look at an example.
::让我们举个例子。

Using the unit scale above, what would be the relationship between the scale dimensions and the actual dimensions for an object 24 feet long?
::使用以上单位比例尺,一个物体24英尺长的尺度尺寸与实际尺寸之间的关系是什么?

First, let’s think about your unit scale.
::首先,让我们想想你的单位规模。

$\begin{aligned} 1 inch = 3 feet \end{aligned}$
::1英寸=3英尺

Next, use the unit scale to write a proportion of the scale dimensions to the actual dimensions.
::其次,使用单位比例尺将比例尺维度与实际维度成比例。

$\begin{aligned} \frac{1 in}{3 ft} = \frac{x in}{24 ft} \end{aligned}$
Then, cross multiply.
::1 英寸 3 ft=x in24, 乘以乘以乘以。

$\begin{aligned} \begin{array}{rcl} \frac{1 in}{3 ft} & = & \frac{x in}{24 ft} \\ 3 x & = & 1 \times 24 \\ 3 x & = & 24 \end{array} \end{aligned}$

::1 英寸 3 ft=x in24 ft3x= 1x243x=24

Then, divide both sides by 3 to solve for $\begin{aligned} x \end{aligned}$ .
::然后把两边除以3 解决x

$\begin{aligned} \begin{array}{rcl} 3 x & = & 24 \\ \frac{3 x}{3} & = & \frac{24}{3} \\ x & = & 8 \end{array} \end{aligned}$

::3x=243x3=243x=8

The answer is 8.
::答案是8岁

Therefore , $\begin{aligned} 8 inches = 24 feet \end{aligned}$ .
::因此,8英寸=24英尺。

If you know the scale dimensions and the unit scale, then you can find the actual dimensions with the unit scale.
::如果您知道规模尺寸和单位尺度,那么您就可以找到单位尺度的实际尺寸。

If you know the actual dimensions and the unit scale, then you can find the scale dimensions with the unit scale.
::如果您知道实际尺寸和单位尺度,那么您就可以用单位尺度找到尺度尺寸。

Let’s look at an example.
::让我们举个例子。

What is the scale length of the object if the unit scale is 2 inches : 4 feet and the actual dimensions of the object is 20 feet?
::如果单位大小为2英寸:4英尺,而物体的实际尺寸为20英尺,那么物体的尺度长度是多少?

First, let’s think about your scale.
::首先,让我们想想你的规模。

$\begin{aligned} 2 inches = 4 feet \end{aligned}$
::2英寸=4英尺

Next, use the scale dimensions to write a proportion of the scale dimensions to the actual dimensions.
::其次,使用比例尺维度将比例尺维度与实际维度成比例。

$\begin{aligned} \frac{2 in}{4 ft} = \frac{x in}{20 ft} \end{aligned}$

::2 英寸 4 英尺=x in20 英尺

Then, cross multiply.
::然后,交叉倍增。

$\begin{aligned} \begin{array}{rcl} \frac{2 in}{4 ft} & = & \frac{x in}{20 ft} \\ 4 x & = & 2 \times 20 \\ 4 x & = & 40 \end{array} \end{aligned}$

::2 英寸 4 英尺=x 20 英尺4x=2xxx20 ×204x=40

Then, divide both sides by 4 to solve for $\begin{aligned} x \end{aligned}$ .
$\begin{aligned} \begin{array}{rcl} 4 x & = & 40 \\ \frac{4 x}{4} & = & \frac{40}{4} \\ x & = & 10 \end{array} \end{aligned}$
The answer is 10.
::然后,将两边除以 4 来解析 x. 4x= 404x4= 404x= 10 答案是 10。

Therefore, $\begin{aligned} 10 inches = 20 feet \end{aligned}$ .
::因此,10英寸等于20英尺

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Josh’s mountainous model.
::更早之前, Josh的山地模型给您带来了问题。

Josh is using scale dimensions of $\begin{aligned} 1 in = 2000 ft \end{aligned}$ . Mount Everest is 29,035 feet.
::Josh正在使用2000年1月1日的尺度尺寸。珠穆朗玛峰是29,035英尺。

First, let’s think about your unit scale.
::首先,让我们想想你的单位规模。

$\begin{aligned} 1 inch = 2000 feet \end{aligned}$
::1英寸=2000英尺

Next, use the unit scale to write a proportion of the scale dimensions to the actual dimensions.
::其次,使用单位比例尺将比例尺维度与实际维度成比例。

$\begin{aligned} \frac{1 in}{2000 ft} = \frac{x in}{29035 ft} \end{aligned}$

::2000年 1 吨=x 29035 吨

Then, cross multiply.
::然后,交叉倍增。

$\begin{aligned} \begin{array}{rcl} \frac{1 in}{2000 ft} & = & \frac{x in}{29, 035 ft} \\ 2000 x & = & 1 \times 29, 035 \\ 2000 x & = & 29, 035 \end{array} \end{aligned}$

::在2000年, 1 ft=x in29, 035 ft2000x=1x29, 0352000x=29, 035x=29, 035

Then, divide both sides by 2000 to solve for $\begin{aligned} x \end{aligned}$ .
::然后在2000年前把两边分开解决x的问题

$\begin{aligned} \begin{array}{rcl} 2000 x & = & 29, 035 \\ \frac{2000 x}{2000} & = & \frac{29, 035}{2000} \\ x & = & 14.5 \end{array} \end{aligned}$

::2000x=29,0352000x2000=29,0352000x=14.5

The answer is 14.5.
::答案是14.5

Josh’s model of Mount Everest would be 14.5 inches tall.
::Josh的珠穆朗玛峰模型将高达14.5英寸。

Example 2
::例2

Using a unit scale of 1 inch : 8 feet, what is the actual dimension of an object with a scale dimension for length of 5 inches?
::使用 1 英寸 : 8 英尺的单位比例尺, 具有 5 英寸长度 5 英寸比例尺大小的物体的实际尺寸是什么 ?

First, let’s think about your unit scale.
::首先,让我们想想你的单位规模。

$\begin{aligned} 1 inch = 8 feet \end{aligned}$
::1英寸=8英尺

Next, use the unit scale to write a proportion of the scale dimensions to the actual dimensions.
::其次,使用单位比例尺将比例尺维度与实际维度成比例。

$\begin{aligned} \frac{1 in}{8 ft} = \frac{5 in}{x ft} \end{aligned}$
Then, cross multiply.
::1 in8 ft=5 inx ftText, 乘以乘数。

$\begin{aligned} \begin{array}{rcl} \frac{1 in}{8 ft} & = & \frac{5 in}{x ft} \\ 1 x & = & 5 \times 8 \\ x & = & 40 \end{array} \end{aligned}$
The answer is 40.
::1 英寸 8 ft= 5 inx ft1x= 5x8x= 40 答案是 40 。

Therefore, the actual dimensions are $\begin{aligned} 5 inches = 40 feet \end{aligned}$ .
::因此,实际尺寸为5英寸=40英尺。

Example 3
::例3

Using a unit scale of 1 inch : 5 feet, what is the actual dimension of an object with a scale dimension for length of 25 feet?
::使用 1 英寸的单位比例尺: 5 英尺, 25 英尺的大小大小物体的实际尺寸是多少?

First, let’s think about your unit scale.
::首先,让我们想想你的单位规模。

$\begin{aligned} 1 inch = 5 feet \end{aligned}$
::1英寸=5英尺

Next, use the unit scale to write a proportion of the scale dimensions to the actual dimensions.
::其次,使用单位比例尺将比例尺维度与实际维度成比例。

$\begin{aligned} \frac{1 in}{5 ft} = \frac{x in}{25 ft} \end{aligned}$
Then, cross multiply.
::1 英寸 5 ft=x in25, 乘以乘以25, 乘以乘以。

$\begin{aligned} \begin{array}{rcl} \frac{1 in}{5 ft} & = & \frac{x in}{25 ft} \\ 5 x & = & 1 \times 25 \\ 5 x & = & 25 \end{array} \end{aligned}$
Then, divide both sides by 5 to solve for $\begin{aligned} x \end{aligned}$ .
::1 英寸 5 ft=x in 25 ft5x= 1xxxxxx= 25, 然后将两边除以 5 以解析 x 。

$\begin{aligned} \begin{array}{rcl} 5 x & = & 25 \\ \frac{5 x}{5} & = & \frac{25}{5} \\ x & = & 5 \end{array} \end{aligned}$

::5x=2555x5=2555x5=5

The answer is 5.
::答案是5。

The actual dimensions are $\begin{aligned} 5 inches = 25 feet \end{aligned}$ .
::实际尺寸为5英寸=25英尺。

Example 4
::例4

Using a unit scale of 1 inch : 5 feet, what is the actual dimension of an object with a scale dimension for length of 3 inches?
::使用 1 英寸的单位比例尺 : 5 英尺, 具有 3 英寸长度 3 英寸比例尺大小的物体的实际尺寸是什么 ?

First, let’s think about your unit scale.
::首先,让我们想想你的单位规模。

$\begin{aligned} 1 inch = 5 feet \end{aligned}$
::1英寸=5英尺

Next, use the unit scale to write a proportion of the scale dimensions to the actual dimensions.
::其次,使用单位比例尺将比例尺维度与实际维度成比例。

$\begin{aligned} \frac{1 in}{5 ft} = \frac{3 in}{x ft} \end{aligned}$

::1 英寸 5 英尺= 3 英寸平方英寸

Then, cross multiply.
::然后,交叉倍增。

$\begin{aligned} \begin{array}{rcl} \frac{1 in}{5 ft} & = & \frac{3 in}{x ft} \\ 1 x & = & 3 \times 5 \\ x & = & 15 \end{array} \end{aligned}$

The answer is 15.
::1 英寸 5 ft= 3 inx ft1x= 3x5x= 15 答案是 15 。

The actual dimensions are $\begin{aligned} 3 inches = 15 feet \end{aligned}$ .
::实际尺寸是3英寸=15英尺

Example 5
::例5

Using a unit scale of 1 inch : 5 feet, what is the actual dimension of an object with a scale dimension for length of 75 feet?
::使用1英寸的单位比例尺:1英寸:5英尺,一个具有75英尺长度的尺度尺寸物体的实际尺寸是多少?

First, let’s think about your unit scale.
::首先,让我们想想你的单位规模。

$\begin{aligned} 1 inch = 5 feet \end{aligned}$
::1英寸=5英尺

Next, use the unit scale to write a proportion of the scale dimensions to the actual dimensions.
::其次,使用单位比例尺将比例尺维度与实际维度成比例。

$\begin{aligned} \frac{1 in}{5 ft} = \frac{x in}{75 ft} \end{aligned}$

::1 英寸 5 ft=x in75 ft

Then, cross multiply.
::然后,交叉倍增。

$\begin{aligned} \begin{array}{rcl} \frac{1 in}{5 ft} & = & \frac{x in}{75 ft} \\ 5 x & = & 1 \times 75 \\ 5 x & = & 75 \end{array} \end{aligned}$

::1 英寸 5 ft=x in75 ft5x=1x755x=75

Then, divide both sides by 5 to solve for $\begin{aligned} x \end{aligned}$ .
::然后把两边除以5 解决x

$\begin{aligned} \begin{array}{rcl} 5 x & = & 75 \\ \frac{5 x}{5} & = & \frac{75}{5} \\ x & = & 15 \end{array} \end{aligned}$

::5x=755x5=755x15

The answer is 15.
::答案是15岁

The actual dimensions are $\begin{aligned} 15 inches = 75 feet \end{aligned}$ .
::实际尺寸为15英寸=75英尺。

Review
::回顾

Find the length of the scale dimension given the scale proportion. Write a proportion and an answer for each problem. There are two answers for each problem.
::查找比例尺大小的长度。给每个问题写一个比例和答案。每个问题有两个答案。

1. Scale is $\begin{aligned} 1^{''} = 2 ft \end{aligned}$ , the actual dimension is 18 feet
::1. 比例为12英尺,实际尺寸为18英尺

2. Scale is $\begin{aligned} 1^{''} = 5 feet \end{aligned}$ , the actual dimension is 20 feet
::2. 面积为15英尺,实际尺寸为20英尺

3. Scale is $\begin{aligned} 1^{''} = 2 feet \end{aligned}$ , actual dimension is 10 feet
::3. 面积为12英尺,实际尺寸为10英尺

4. Scale is $\begin{aligned} 1^{''} = 12 feet \end{aligned}$ , actual dimension is 72 feet
::4. 面积为112英尺,实际尺寸为72英尺

5. Scale is $\begin{aligned} 3^{''} = 4 feet \end{aligned}$ , actual dimension is 16 feet
::5. 面积为34英尺,实际尺寸为16英尺

Using a scale of 1 to 2, figure out the actual dimensions given each scale.
::使用1至2的尺度,找出每个尺度的实际尺寸。

6. 4 to $\begin{aligned} \underline{} \end{aligned}$
::6. 4至_

7. 6 to $\begin{aligned} \underline{} \end{aligned}$
::7. 6至_

8. 9 to $\begin{aligned} \underline{} \end{aligned}$
::8. 9至_______________________________________________________________________________________________________________

9. 12 to $\begin{aligned} \underline{} \end{aligned}$
::9. 12至_

10. 14 to $\begin{aligned} \underline{} \end{aligned}$
::10. 14至_

Using a scale of 3 to 4, figure out the actual dimensions given each scale.
::使用3至4的尺度,找出每个尺度的实际尺寸。

11. 6 to $\begin{aligned} \underline{} \end{aligned}$
::11. 6至_

12. 9 to $\begin{aligned} \underline{} \end{aligned}$
::12时9分至_

13. 12 to $\begin{aligned} \underline{} \end{aligned}$
::13. 12至_

14. 18 to $\begin{aligned} \underline{} \end{aligned}$
::14. 18至_________________________________________________________________________________________________________________________________________________________________________________________________________

15. 36 to $\begin{aligned} \underline{} \end{aligned}$
::15. 36至_

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

章节大纲

Unit Scale ::单位比例缩缩

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

Unit Scale
::单位比例缩缩

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