课程： 1.18 立体模型 | The Way To Learn

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Cucc 模型模型

How do you write a math problem to represent a physical object?
::您如何写出数学问题来代表物理物体 ?

The video below, and the first portion of the lesson thereafter, both describe different challenges involving modeling a box created by cutting corners out of a flat sheet of cardboard and then folding up the sides. In the video, the specific problem is to identify the size of the cardboard needed to result in a given volume . In the lesson, the challenge is to identify the greatest volume possible from a given size sheet of cardboard.
::下面的视频以及随后的第一部分课程,都描述了不同的挑战,包括将纸板平板上的角切开而形成的盒子建模,然后折叠两边。在视频中,具体的问题是确定纸板的大小,以便形成一定的体积。在课程中,挑战是如何从纸板的某一体积中找出尽可能多的体积。

How would you mathematically model a divider to split the volume of the box described in the lesson into multiple spaces?
::您在数学上如何建模分隔符, 将课中描述的框体积分成多个空格 ?

Cubic Models
::Cucc 模型模型

Cubic functions and piece-wise functions can be used to model real-world situations, allowing you to identify missing bits of information you may need to complete a project. Cubic functions are commonly used to model three-dimensional objects to allow you to identify a missing dimension or explore the result of changes to one or more dimensions .
::立方函数和计件函数可用于模拟真实世界状况, 从而可以识别完成工程可能需要的缺失信息位数。立方函数通常用于模拟三维天体, 以便您识别缺失的维度或探索一个或一个以上维值变化的结果。

Piece-wise functions may be used to model the interactions of multiple items each previously modeled by a simpler function .
::可使用小数函数来模拟多个项目之间的相互作用,每个项目以前都用简单的函数来模拟。

Examples
::实例

Example 1
::例1

Earlier, you were asked to create a different model describing the cardboard box.
::早些时候,你被要求创建一个不同的模型描述纸箱。

There are a number of different possibilities, for instance:
::存在若干不同的可能性,例如:

In the case of the box in the video above: $\begin{aligned} t (v) = v / h \end{aligned}$ could be used to describe the area of the top of the box as a function of the volume.
::在以上视频中的框中:t(v)=v/h可以用来描述框顶部的面积,作为音量的函数。

In the case of the box in the lesson text: $\begin{aligned} d (x) = (12 - x) x \end{aligned}$ or $\begin{aligned} d (x) = (8 - x) x \end{aligned}$ could be used to find the area of a divider run the long way or the wide way across the box as a function of the length of one side of the square corner cut-outs.
::课文中的框:d(x)=(12-x)x 或d(x)=(8-x)x 可用于发现分隔符的面积在横跨框的长途或宽道之间,视角角截断处一面的长度而定。

Example 2
::例2

Consider a situation in which a rectangular piece of cardboard is folded into a box. The folding is made possible by cutting squares out of the four corners of the cardboard.
::考虑一下纸板的长方形片被折叠成盒子的情况。折叠是通过切开纸板四角的方形而得以实现的。

Calculate the maximum volume possible of a box made from a sheet of cardboard 12" x 8".
::计算用纸板 12" x 8" 纸板制成的盒子的最大容量。

The function $\begin{aligned} v (x) = (12 - 2 x) (8 - 2 x) x \end{aligned}$ could be used to represent the volume of the box as a function of x , the side-length of the squares cut out of the corners. If we multiply out the factors of this function we can verify that this is a cubic function:
::函数 v (x) = (12) - 2x (8 - 2x) x 可用于代表框的音量, 以 x 的函数表示角切开的方形的侧长。如果我们乘出此函数的因子, 我们就可以核实这是立方函数 :

			$\begin{aligned} v (x) = (12 - 2 x) (8 - 2 x) x \end{aligned}$ ==> $\begin{aligned} 4 x^{3} - 40 x^{2} + 96 x \end{aligned}$

This function could be used to find the maximum possible volume of the box. We can also analyze the graph to understand how the volume changes as a function of x .
::此函数可用于查找框的最大可能的体积。我们还可以分析图形,了解音量如何变化为 x 的函数。

When analyzing the function to determine the maximum volume of the box, we only look at the portion of the graph that looks “parabolic”. This is because the function ceases to model the situation if x is more than 4. If we cut out {4x4} squares, we would cut out the entire short side of the cardboard rectangle, and we would not be able to make a box. Focusing then on the interval (0, 4) we can see that the volume of the box increases, and then decreases.
::当分析函数以确定框的最大体积时,我们只看图表中显示“抛物线”的部分。这是因为如果 x 大于 4 , 函数停止模拟情况。如果我们切除 {4x4} 方形, 我们就会切除纸板矩形的整个短侧, 无法创建框。然后聚焦于间隔( 0, 4) , 我们可以看到框的体积会增加, 然后减少。

If we are using a graphing calculator and want to know the volume of a box with particular dimensions, we can trace on the graph, input values into the table, or take advantage of the graph being in trace mode. That is, if you press GRAPH to view the graph, and then press TRACE, you can input x values. For example, say that you wanted to cut out squares of side-length 2.5. Press TRACE, then press 2.5, then press ENTER. At the bottom of the screen you will see x = 2.5 and y = 52.5. This tells you that the volume of the box will be 52.5in ³ .
::如果我们使用图形计算器,想要知道一个带有特定维度的盒子的体积, 我们可以在图表上追踪, 输入到表格中, 或者利用图表的跟踪模式。也就是说, 如果您按 GRAPH 查看图形, 然后按 TRACE 键, 您可以输入x 值。例如, 说您想要切除侧长2.5 的方形。按 TRACE, 然后按 2.5, 然后按 ENTER 键。在屏幕底部您可以看到 x = 2.5 和 y = 52. 5 。这表明框的体积将是 52.5 英寸。

Example 3
::例3

Using the information given in Example 2 above, calculate the size of the square corners to cut out to result specifically in a volume of 50 in ³ .
::使用上文例2中提供的信息,计算要切开的广场角的大小,以具体得出50英寸3的体积。

One way to determine the value of x is to graph the constant function y = 50, and find the point where the volume function intersects y = 50. Press Y= and enter 50 into Y2 . Now press GRAPH. You should see the horizontal line y = 50 intersecting the volume function in several places. We are interested in the two intersection points in the interval (0, 4).
::确定 x 值的一个方法就是绘制常数函数 y = 50, 并找到音量函数交叉 y = 50 的点。按 Y = 并输入 50 到 Y2 。现在按 GRAPH 键。您应该看到水平线 y = 50 在多个地方交叉音量函数。我们感兴趣的是间隔( 0, 4) 中的两个交叉点。

To find a good approximation of an intersection point, trace close to one of the two points. If you trace close to the first point, you will see that it is around x = 0.8. Now press 2 ^nd , CALC, and choose option 5, INTERSECT. The calculator will send you back to the graph screen, and ask you to choose the first curve. (The calculator does this in case you have more than two functions graphed at the same time.) You should see the cubic equation at the top of the screen. Press ENTER, and the calculator will ask you for the second curve. You should see y = 50 at the top of the screen. Press ENTER, and then enter a guess. (If you have already traced close to an intersection point, and you only have two functions graphed, you can simply press ENTER three times.) You should now see the coordinates of the intersection point at the bottom of the screen: x is approximately 0.723 . (You can use the same process to estimate the other intersection point.)
::要找到一个交叉点的近似点, 请追踪到两个点中的一个点。如果您跟踪到第一个点附近, 您将会看到它是在 x = 0. 8 左右。现在按 CALC, 并选择选项 5 , InterSECT 。计算器会将您送回图形屏幕, 并请求您选择第一个曲线。 (计算器这样做, 以防您同时有超过两个功能。) 您应该看到屏幕顶部的立方程。按 ENTER 键, 计算器会要求您选择第二个曲线。您应该在屏幕顶部看到 y = 50 。按 ENTER 键, 然后输入一个猜测。 ( 如果您已经追踪到一个交叉点, 并且您只有两个函数图形, 您可以简单地按 ENTER 三次。) 您现在应该看到屏幕底部的交叉点的坐标 : x 是 0. 0. 723 。 ( 您可以使用相同的进程来估计另一个交叉点 ) 。

A cubic function can be used to model situations that involve volume, but can also be used to model situations that follow particular growth patterns.
::立方函数可用于模拟涉及数量的情况,但也可用于模拟遵循特定增长模式的情况。

Example 4
::例4

Piece-wise functions can be used to describe situations in which quantitative relationships are different in different intervals within the domain of the function. For example, consider a situation in which a wireless provider offers customers a monthly plan that costs $50, but then charges $0.40 cents per minute for every minute over 1000 included daytime minutes.
::零星函数可以用来描述数量关系在函数范围内不同时间间隔不同的情况。例如,可以考虑无线服务提供商向客户提供每月50美元的月计划,但随后每分钟每分钟收费0.40美分,超过1000分钟,包括白天分钟。

Model the monthly cost, C, of the plan as a function of m, the number of daytime minutes you use:
::以计划每月成本C为模型,以 m 函数,即你使用的日间分钟数为函数:

\begin{aligned} C (m) = {\begin{cases} 50, & m \leq 1000 \\ 50 + 0.40 (m - 1000), & m > 1000 \end{cases} \end{aligned}

This function is comprised of a constant function, and a linear function with 0.40. If in a given month you use 1000 minutes or fewer, your monthly cost is a constant $50. If you use more than 1000, each additional minute influences the value of C. For example, if you use 1,020 minutes, your cost is:
::此函数由不变函数和0.40的线性函数组成。如果在一个特定月份中您使用1000分钟或更少,则每月费用为50美元不变值。如果使用1000以上,则每增加一分钟就会影响C值。例如,如果使用1,020分钟,您的费用为:

		C (1020) = 50 + 0.40(1020 - 1000)
		$\begin{aligned} = \end{aligned}$ 50 + 0.4(20)
		$\begin{aligned} = \end{aligned}$ 50 + 8
		$\begin{aligned} = \end{aligned}$ $58.00

It is important to note that in this kind of situation, the time used may to be rounded to the nearest minute. So, for example, if you use 20.5 minutes, you will be charged for 21 minutes. This is an example of a non- continuous , or discontinuous, function, where there are definite steps from value to value rather than a smooth line connecting all possible values.
::必须指出,在这种情形下,使用的时间可以四舍五入到最接近的分钟。例如,如果使用20.5分钟,您将被收费21分钟。这是一个非连续或不连续功能的例子,从价值到价值都有明确的步骤,而不是连接所有可能值的平滑线。

Example 5
::例5

The profits for a business can be determined by subtracting the costs from the revenue . Suppose the revenue of a business is modeled by the function $\begin{aligned} R (x) = 5 x - 0.01 x^{2} \end{aligned}$ , and the costs of manufacturing the product is modeled by $\begin{aligned} C (x) = 100 + 2 x \end{aligned}$ , where x is the number of units of the product.
::企业的利润可以通过从收入中减去成本来确定。假设企业的收入以R(x)=5x-0.01x2函数为模型,而产品的制造成本以C(x)=100+2x为模型,其中x是产品单位数。

Write a function P ( x ) to model the company’s profits.
::撰写函数 P(x) 来模拟公司的利润。

P ( x ) = R ( x ) - C ( x ) = -0.01 x ² + 3 x - 100
::P(x) = R(x) - C(x) = - 0.01x2 + 3x - 100

Graph P ( x ) and determine the maximum profit.
::图P(x)和确定最大利润。

The maximum profit is 125 (usually in thousands, or another larger unit!)
::最大利润125(通常以千计, 或以另一大单位! )

Example 6
::例6

Express the following situation as a : You are running a small business making wooden jewelry boxes. It costs you $5.00 per unit to produce wooden boxes, plus an initial investment of $300 in other materials. It then costs you an additional $2.00 per box to decorate the boxes.
::将以下情形表述为 : 您正在经营一个小生意, 做木首饰箱。生产木箱每单位需要5.00美元, 加上最初对其他材料投资300美元。然后您需要每箱额外2.00美元来装饰盒子。

Initial cost function: C ₁ ( x ) = 5 x + 300
::初始成本功能:C1(x)=5x+300

Second cost function C ₂ ( x ) = 2 x
::第二个成本函数C2(x)=2x

Composition: C( x ) = C ₁ (C ₂ ( x )) = 5(2 x ) + 300 = 10 x + 300
::组成:C(x) = C1 (C2(x) = 5(2x) + 300 = 10x + 300

Review
::回顾

A box is to be made by cutting squares out of the corners of a rectangular piece of cardboard. The dimensions of the cardboard are n inches by m inches. Assume that n > m . a. Write a model for the volume of the box. b. What is the largest square that can be cut out of the corners of the cardboard?
::框须通过切开纸板矩形片角角的方形来制作。纸板的尺寸是每英寸 n 英寸。假设 n > m. a. , 写一个框体积的模型。 b. 纸板角中最大的方形是什么?

Suppose you were told that you could use a single sheet of paper for notes on your math final. The instructor says that you may use any sheet you like, but the shape must be a quadrangle and the perimeter may not exceed 45in. a) What dimensions should you use to provide the greatest area for your notes? b) How does a graphing calculator help to simplify this question?
::假设有人告诉您, 您数学期末考试的笔记可以使用一张纸。教官说, 您可以使用任何您喜欢的笔记, 但形状必须是一个矩形, 周界不能超过 45 英寸。 a) 您应该使用什么维度来提供您笔记的最大区域 ? b) 图形计算器如何帮助简化这一问题 ?

Is $\begin{aligned} f (x) = - 3^{4 x} \end{aligned}$ a power function?
::f( x)\\\\\\\\\34x 是一个功率函数吗 ?

Is $\begin{aligned} f (x) = \sqrt[3]{8 x^{5}} \end{aligned}$ a power function?
::f( x) = 8x53 是一个功率函数吗 ?

Is $\begin{aligned} g (x) = 7 \cdot 2^{x} \end{aligned}$ a power function? If not, why not?
::g(x) = 72x 是一个功率函数吗? 如果没有,为什么不呢?

Is $\begin{aligned} h (x) = 2 x^{- 5} \end{aligned}$ a power function? If not, why not?
::h(x)=2x-5 是一个功率函数吗? 如果没有,为什么不呢?

The volume v of a sphere varies directly as the cube of the radius r . When the radius of a sphere is 6 cm, the volume is $\begin{aligned} 904.779 c m^{3} \end{aligned}$ . What is the radius of a sphere whose volume is $\begin{aligned} 268.083 c m^{3} \end{aligned}$ ?
::当一个球的半径为 6 厘米时, 体积为 904. 779 厘米。体积为 268. 083 厘米的球体的半径是 268. 83 厘米?

The force of gravity (F) acting on an object is inversely proportional to the square of the distance d from the object to the center of the earth. Write an equation that models this situation.
::重力(F)在物体上作用的强度与从物体到地球中心的距离的正方形成反比。写一个公式来模拟这种情况。

Sue and Betty gathered the data in the table below using a 100-watt light bulb and a Calculator-Based Laboratory(CBL) with a light-intensity probe.
::Sue和Betty用100瓦灯泡和以计算器为基础的实验室(CBL)和光强度探测器收集了下表的数据。

Light Intensity Data for a 100w light bulb
::100w光灯泡的光强度数据

Distance (m)	Intensity $\begin{aligned} (W / m^{2}) \end{aligned}$
1.0	7.95
1.5	3.53
2.0	2.01
2.5	1.27
3.0	0.90

Use your calculator to find the power regression model of the data.
::使用您的计算器查找数据的功率回归模型。

Describe the relationship between the intensity and distance modeled with the equation in Q #9
::描述与Q#9中方程模型的强度和距离之间的关系

Use the regression model from Q #9 to predict the intensity of an object 2.75 meters away.
::使用Q#9的回归模型来预测一个物体在2.75米外的强度

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。

章节大纲

常规

Cucc 模型模型

Cubic Models ::Cucc 模型模型

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Example 6 ::例6

Review ::回顾

Review (Answers) ::回顾(答复)