课程： 9.6 密度问题-interactive

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人口密度问题

Density is a ratio of two measurements of an object. In the past you might have seen the following formula for density:
::密度是一个对象的两度测量比率。过去你可能见过以下密度公式:

$\begin{aligned} d = \frac{m}{V} \end{aligned}$
::d=mV = mV = mV =mV

This formula states that the density of an object is the ratio of its mass to its . If two objects have the same volume, but different masses, the object with the greater mass will be denser.
::此公式表示一个对象的密度是其质量与其质量之比。如果两个对象的体积相同,但质量不同,则质量较大的对象的密度会更高。

You can also think about the density of an area of land such as a city. In this case, you would look at the ratio of number of people to area of land.
::您也可以想一想土地面积的密度,例如城市。在此情况下,您可以查看人口与土地面积的比例。

$\begin{aligned} population density = \frac{number of people}{area of land} \end{aligned}$
::人口密度=土地人口面积数

A city will have a higher population density than land in the country, because more people live in each square mile of a city than live in each square mile out in the country.
::一个城市的人口密度将高于该国的土地密度,因为生活在城市每一平方英里的人口多于居住在该国每一平方英里的人口。

In general, density is a ratio of the mass/people/things to volume/area. Given the density and one of the two quantities in the ratio, you can solve for the other quantity in the ratio.
::一般而言,密度是质量/人/物与体积/面积之比。根据密度和两个数量中的一个,您可以解决该比率中的其他数量。

Density
::密度

Adjust the slider to change the population. Observe how the population density changes as the population increases or decreases.
::调整滑块以改变人口。观察人口密度随人口增加或减少而变化。

Let's look at a few problems involving density.
::让我们来研究一下与密度有关的一些问题。

1. Different metals have different densities. The density of gold is 19.3 grams per $\begin{aligned} {cm}^{3} \end{aligned}$ , the density of copper is 8.86 grams per $\begin{aligned} {cm}^{3} \end{aligned}$ , and the density of bronze is 9.87 grams per $\begin{aligned} {cm}^{3} \end{aligned}$ . Your bracelet is 106 grams and has a volume of 12 cubic centimeters. What metal is your bracelet made out of?
::1. 不同金属的密度不同:金的密度为每厘米19.3克,铜的密度为每厘米8.86克,铜的密度为每厘米9.87克,铜的密度为每厘米9.87克。你的手镯为106克,体积为12立方厘米。你的手镯是用什么金属做的?

The density of your bracelet is $\begin{aligned} d = \frac{m}{V} = \frac{106}{12} \approx 8.83 grams per {cm}^{3} \end{aligned}$ . It is probably copper.
::您手镯的密度是 d=mV=106128.83克/厘米。它可能是铜。

2. There are approximately 17,620 people per square mile in New York City. If New York City is 468 square miles, approximately how many people live in New York City?
::2. 纽约市每平方英里约有17 620人,纽约市每平方英里468平方英里,约有多少人住在纽约市?

Map of New York City showing population density distribution with people icons.

17,620 is the population density of New York City.
::17 620是纽约市的人口密度。

$\begin{aligned} population density & = \frac{number of people}{area of land} \\ 17, 620 people per square mile & = \frac{number of people}{468 {mi}^{2}} \end{aligned}$
::人口密度=土地面积人口数量17 620人/平方英里=468 MI2人/平方英里

Solve this equation by multiplying both sides by 468. New York City has approximately 8,246,160 people.
::将双方乘以468. 纽约市约有8 246 160人。

3. Helium is a gas lighter than air that is used in balloons. The density of helium is approximately 0.18 kilograms per cubic meter. You get 16 helium balloons for your 16th birthday, each with a volume of 0.15 cubic meters. How many kilograms of helium do the balloons contain?
::3. 氦是比气球中所使用的空气更轻的气体,其密度约为每立方米0.18公斤,在16岁生日时,每16个气球的体积为0.15立方米,得到16个氢气球。气球内含多少公斤的氦?

The 16 balloons together have a volume of $\begin{aligned} 16 (0.15) = 2.4 m^{3} \end{aligned}$ . Each cubic meter has a mass of 0.18 kg, so the balloons contain
::16个气球共16个气球的体积为16(0.15)=2.4立方米。每立方米的重量为0.18公斤,因此气球含有

$\begin{aligned} M = d V = (0.18) (2.4) = 0.432 kilograms of helium \end{aligned}$ .
::M=dV=(0.18)(2.4)=0.432公斤。

Examples
::实例实例实例实例

Example 1
::例1

A balloon and a rock each have a volume of 1 cubic foot. The rock is much heavier than the balloon. What does this say about the density of the balloon compared to the density of the rock?
::气球和岩石各有1立方英尺的体积。岩石比气球重得多。与岩石密度相比, 气球的密度是多少?

Because the balloon and the rock each have the same volume, the object with the greater mass has the greater density. Therefore, the rock is more dense than the balloon.
::由于气球和岩石的体积相同,质量较大的物体的密度更高。因此,岩石比气球的密度更高。

Example 2
::例2

A cylindrical candle with a radius of 3 centimeters and a height of 8 centimeters has a mass of 300 grams.
::圆柱形蜡烛半径为3厘米,高度为8厘米,质量为300克。

i) What is the density of the candle?
:一) 蜡烛的密度是多少?

$\begin{aligned} V_{cylinder} & = π r^{2} h = π (3)^{2} (8) = 226.2 {cm}^{3} \\ d & = \frac{m}{V_{cylinder}} = \frac{300}{226.2} \\ d & = 1.33 grams per cubic centimeter \end{aligned}$
::Vcylinderr2h(3)(2)(8)=226.2cm3d=mVcinder=300226.2d=1.33克/每立方厘米

ii) Another candle made out of the same wax is formed into a cube. The mass of this candle is 475 grams. What are the dimensions of the cube?
:二) 另一根由同一蜡制成的蜡烛组成立方体,其质量为475克。

$\begin{aligned} d & = \frac{m}{V_{cube}} so 1.33 = \frac{475}{V_{cube}} \\ V_{cube} & = \frac{475}{1.33} = 357.14 {cm}^{3} \end{aligned}$
::d=mVcube, 因此1.33=475VcubeVcube=4751.33=357.14立方厘米

Because the candle is a cube, all of its edges must be the same length. If $\begin{aligned} s \end{aligned}$ is the length of each edge of the cube, then
::因为蜡烛是立方体, 其所有边缘的长度必须相同。如果秒是立方体每个边缘的长度, 那么

$\begin{aligned} V_{cube} = s^{3} & = 357.14 \\ s & \approx 7.1 cm \end{aligned}$
::立方体=s3=357.14s=7.1厘米

iii) Another candle made out of the same wax is formed into a cone. The diameter of the base of the cone is 4 centimeters and the height of the cone is 12 centimeters. What is the mass of the cone candle?
:三) 由同一蜡制成的另一根蜡烛形成锥体,锥体的直径为4厘米,锥体的高度为12厘米。

$\begin{aligned} V_{cone} & = \frac{π r^{2} h}{3} = \frac{π (2^{2}) (12)}{3} \approx 50.27 {cm}^{3} \\ d & = \frac{m}{V_{cone}} so m = d V_{cone} \\ m & = (1.33) (50.27) = 66.85 grams \end{aligned}$ .
::Vconer2h3(22)(12)350.27cm3d=mVcone so m=dVconem=(1.33)(50.27)=66.85克。

CK-12 PLIX Interactive
::CK-12 PLIX 交互式互动

Summary

The formula for density is $\begin{aligned} d = \frac{m}{V} \end{aligned}$
The population density can be found with $\begin{aligned} population density = \frac{number of people}{area of land} \end{aligned}$
::密度的公式是 d=mV 人口密度可与人口密度=土地人口面积数同时找到。

Review
::审查审查审查审查

1. 30 students return to the classroom after playing outside at recess. Over the course of the next hour, each person produces 800 BTUs (British Thermal Units). The room is 20 feet by 20 feet by 12 feet. How many BTUs per cubic foot were produced?
::1. 30名学生在课外休息后返回教室,在接下来的一小时里,每人生产800个BTU(英国热量单位),房间20英尺乘20英尺乘12英尺,每立方英尺生产多少BTU?

At $\begin{aligned} 20^{\circ} C \end{aligned}$ , skim milk has a density 1.033 kilograms per liter, while light cream has a density of 1.012 kilograms per liter.
::20°C时,脱脂牛奶密度为每升1.033公斤,轻型奶油密度为每升1.012公斤。

2. At $\begin{aligned} 20^{\circ} C \end{aligned}$ , what is the mass of one gallon of skim milk if 1 liter is approximately 0.264 gallons?
::2. 在20°C,如果1升大约为0.264加仑,1加仑脱脂牛奶的质量是多少?

3. At $\begin{aligned} 20^{\circ} C \end{aligned}$ , a container holding 9 liters of liquid is 9.108 kilograms. Does the container hold skim milk or light cream?
::3. 在20°C时,装有9升液体的容器为9.108公斤,容器装的是滑雪牛奶还是轻奶油?

In Manhattan, there are 15 CoffeeStops per 100,000 people.
::在曼哈顿,每10万人有15个咖啡站。

4. Manhattan has approximately 1,602,000 people. How many CoffeeStops are there?
::4. 曼哈顿约有1 602 000人,有多少个咖啡站?

5. Manhattan is approximately 34 square miles. What is the density of CoffeeStops per square mile?
::5. 曼哈顿约34平方英里,每平方英里咖啡站的密度是多少?

6. On average, each person in Manhattan goes to a CoffeeStop three times a week to buy a cup of coffee. About how many cups of coffee does each Manhattan CoffeeStop serve each day?
::6. 平均而言,曼哈顿每人每周三次到咖啡站买咖啡,每个曼哈顿咖啡站每天要供应多少杯咖啡?

The population density of a certain 750 square mile area in 1980 was 1077 people per square mile. In 1990, the population density was 1137 people per square mile. In 2000, the population density was 1144 people per square mile. In 2010, the population density was 1193 people per square mile.
::1980年,750平方英里的人口密度为每平方英里1 077人,1990年人口密度为每平方英里1 137人,2000年人口密度为每平方英里1 144人,2010年人口密度为每平方英里1 193人。

7. What was the population of this area in 1980?
::7. 1980年该地区人口是多少?

8. How much did the population change between 1980 and 2010?
::8. 1980年至2010年之间人口变化有多大?

9. Which decade had the smallest percentage increase in population?
::9. 哪个十年的人口增长率最小?

The density of gold is $\begin{aligned} 19.3 grams per {cm}^{3} \end{aligned}$ .
::黄金密度为每立方厘米19.3克。

10. What is the mass of a gold block that measures 10 cm by 3 cm by 3 cm?
::10. 代表10厘米乘3厘米乘3厘米的金块质量是多少?

11. A pyramid shaped object is made out of pure gold. The base of the pyramid is a square and the height of the pyramid is 5 centimeters. The mass of the pyramid is 1.5 kilograms. What are the dimensions of the base of the pyramid?
::11. 金字塔形物体由纯金制成,金字塔的底部是方形,金字塔的高度是5厘米,金字塔的重量是1.5公斤,金字塔底部的尺寸是多少?

In general, if one object has a lower density than a liquid, it will float in the liquid.
::一般而言,如果一个物体的密度低于液体,它将漂浮在液体中。

12. At $\begin{aligned} 4^{\circ} C \end{aligned}$ , water has a density of $\begin{aligned} 1000 kg / m^{3} \end{aligned}$ . The density of ice is $\begin{aligned} 917 kg / m^{3} \end{aligned}$ . Why does ice float in water?
::12. 在4C,水的密度为1000千克/立方米,冰的密度为917千克/立方米,为什么冰在水中漂浮?

13. A piece of wood has a mass of 25 kg and a volume of $\begin{aligned} 0.000385 m^{3} \end{aligned}$ . What is the density of the wood?
::13. 木块的重量为25公斤,体积为0.000385立方米。木块的密度是多少?

14. Do you think the wood from #13 will float in water?
::你认为13号的木头会浮在水里吗?

15. A cylinder of lead has a diameter of 3 centimeters and a height of 8 centimeters. The density of lead is $\begin{aligned} 11.4 g / {cm}^{3} \end{aligned}$ . What is the mass of the cylinder?
::15. 铅气瓶直径为3厘米,高度为8厘米,铅的密度为11.4克/立方厘米。

16. What is the relationship between air density and elevation? Sketch a prospective graph of the relationship, with air density on the y-axis and elevation on the x-axis. What sort of function might model this relationship? Research to find data on air density at different elevations, sketch a graph, and try to create a model. Do more research to see if there is an equation which models this relationship (while holding the other variables constant).
::16. 空气密度和高度之间的关系是什么? 绘制一个具有 Y 轴的空气密度和 x 轴的高度的关系前景图。什么样的功能可以模拟这种关系? 研究在不同高度寻找空气密度数据,绘制一个图形,并尝试创建模型。做更多的研究,看看是否存在一个公式来模拟这种关系(同时保持其他变量不变 ) 。

17. Below find listed the population of several states, as well as the land area. Compute the population density for each. What do you think the relationship is between land area and population density? Research to find more data and graph it.
::17. 下文列出了几个州的人口以及土地面积。计算每个州的人口密度。你认为土地面积与人口密度之间的关系是什么?研究寻找更多的数据并绘制图表。

State	Population (millions)	Land Area (thousands of square miles)
New Jersey	9.0	7.4
California	39	160
Louisiana	4.7	43
Texas	27	260
Colorado	5.5	100
Kansas	2.9	82
Alaska	0.74	570

18. The table below shows the percentage of different gases in the air we inhale and the air we exhale, by volume. The tidal volume is the air that goes in or out during a single breath--for normal adults it's a half liter.
::18. 下表按体积显示我们吸入的空气中不同气体的百分比和我们吸入的空气中不同气体的百分比,潮汐体积是正常成年人呼吸时进出的空气,只有半升。

Gas	Inhale	Exhale
Nitrogen	78	78
Oxygen	21	16
Argon	0.90	0.90
Carbon Dioxide	0.0410	4.0

What is the volume of the oxygen inhaled? What's the volume of the oxygen exhaled? What's the volume of the carbon dioxide exhaled? The acceptable EPA limit on carbon monoxide is 50 parts per million. What volume percentage does that represent? If a single breath takes in 50 ppm of carbon monoxide, what is the volume of carbon monoxide in that breath? What is the ppm concentration of carbon dioxide in the air?
::吸入的氧气量是多少? 吸入的氧气量是多少? 吸入的二氧化碳量是多少? 排放的二氧化碳量是多少? 对一氧化碳的可接受环保局限制是百万分之50。这代表了多少比例? 如果呼吸一氧化碳的每百万分之50,一氧化碳的一氧化碳含量是多少? 空气中二氧化碳的含量是多少?

Review (Answers)
::审查(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

人口密度问题

Density ::密度

Examples ::实例实例实例实例

Example 1 ::例1

Example 2 ::例2

CK-12 PLIX Interactive ::CK-12 PLIX 交互式互动

Review ::审查审查审查审查

Review (Answers) ::审查(答复)