## 数学代写|线性代数代写linear algebra代考|MATH1071

2023年1月5日

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## 数学代写|线性代数代写linear algebra代考|Constructing a Nutritious Weight-Loss Diet

The formula for the Cambridge Diet, a popular diet in the $1980 \mathrm{~s}$, was based on years of research. A team of scientists headed by Dr. Alan $\mathrm{H}$. Howard developed this diet at Cambridge University after more than eight years of clinical work with obese patients. ${ }^1$ The very low-calorie powdered formula diet combines a precise balance of carbohydrate, high-quality protein, and fat, together with vitamins, minerals, trace elements, and electrolytes. Millions of persons have used the diet to achieve rapid and substantial weight loss.

To achieve the desired amounts and proportions of nutrients, Dr. Howard had to incorporate a large variety of foodstuffs in the diet. Each foodstuff supplied several of the required ingredients, but not in the correct proportions. For instance, nonfat milk was a major source of protein but contained too much calcium. So soy flour was used for part of the protein because soy flour contains little calcium. However, soy flour contains proportionally too much fat, so whey was added since it supplies less fat in relation to calcium. Unfortunately, whey contains too much carbohydrate….

The following cxample illustrates the problem on a small scalc. Listed in Table 1 are three of the ingredients in the diet, together with the amounts of certain nutrients supplied by 100 grams $(\mathrm{g})$ of each ingredient. ${ }^2$

EXAMPLE 1 If possible, find some combination of nonfat milk, soy flour, and whey to provide the exact amounts of protein, carbohydrate, and fat supplied by the diet in one day (Table 1).

SOLUTION Let $x_1, x_2$, and $x_3$, respectively, denote the number of units (100 $\left.\mathrm{g}\right)$ of these foodstuffs. One approach to the problem is to derive equations for each nutrient separately. For instance, the product
$$\left{\begin{array}{c} x_1 \text { units of } \ \text { nonfat milk } \end{array}\right} \cdot\left{\begin{array}{c} \text { protein per unit } \ \text { of nonfat milk } \end{array}\right}$$
gives the amount of protein supplied by $x_1$ units of nonfat milk. To this amount, we would then add similar products for soy flour and whey and set the resulting sum equal to the amount of protein we need. Analogous calculations would have to be made for each nutrient.

## 数学代写|线性代数代写linear algebra代考|Linear Equations and Electrical Networks

Current flow in a simple electrical network can be described by a system of linear equations. A voltage source such as a battery forces a current of electrons to flow through the network. When the current passes through a resistor (such as a lightbulb or motor), some of the voltage is “used up”; by Ohm’s law, this “voltage drop” across a resistor is given by
$$V=R I$$
where the voltage $V$ is measured in volts, the resistance $R$ in ohms (denoted by $\Omega$ ), and the current flow $I$ in amperes (amps, for short).

The network in Figure 1 contains three closed loops. The currents flowing in loops 1,2 , and 3 are denoted by $I_1, I_2$, and $I_3$, respectively. The designated directions of such loop currents are arbitrary. If a current turns out to be negative, then the actual direction of current flow is opposite to that chosen in the figure. If the current direction shown is away from the positive (longer) side of a battery $(-\mid \mathrm{F})$ around to the negative (shorter) side, the voltage is positive; otherwise, the voltage is negative.
Current flow in a loop is governed by the following rule.

SOLUTION For loop 1 , the current $I_1$ flows through three resistors, and the sum of the $R I$ voltage drops is
$$4 I_1+4 I_1+3 I_1=(4+4+3) I_1=11 I_1$$
Current from loop 2 also flows in part of loop 1 , through the short branch between $A$ and $B$. The associated $R I$ drop there is $3 I_2$ volts. However, the current direction for the branch $A B$ in loop 1 is opposite to that chosen for the flow in loop 2, so the algebraic sum of all $R I$ drops for loop 1 is $11 I_1-3 I_2$. Since the voltage in loop 1 is $+30$ volts, Kirchhoff’s voltage law implies that
$$11 I_1-3 I_2=30$$
The equation for loop 2 is
$$-3 I_1+6 I_2-I_3=5$$
The term $-3 I_1$ comes from the flow of the loop 1 current through the branch $A B$ (with a negative voltage drop because the current flow there is opposite to the flow in loop 2). The term $6 I_2$ is the sum of all resistances in loop 2 , multiplied by the loop current.

# 线性代数代考

## 数学代写|线性代数代写linear algebra代考|Constructing a Nutritious Weight-Loss Diet

\left{\begin{array}{c} x_1 \text { } \ \text { 脱脂牛奶 } \end{array}\right} \cdot\left{\begin{array}{c} \text { 蛋白质每单位 } \ \text { 脱脂牛奶 } \end{array}\right}\left{\begin{array}{c} x_1 \text { } \ \text { 脱脂牛奶 } \end{array}\right} \cdot\left{\begin{array}{c} \text { 蛋白质每单位 } \ \text { 脱脂牛奶 } \end{array}\right}

## 数学代写|线性代数代写linear algebra代考|Linear Equations and Electrical Networks

$$V=R I$$

$$4 I_1+4 I_1+3 I_1=(4+4+3) I_1=11 I_1$$

$$11 I_1-3 I_2=30$$

$$-3 I_1+6 I_2-I_3=5$$

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。