节: 气瓶高地地表面积 | 10.7 圆柱体地表地区高地 | The Way To Learn

The Way To Learn

首页日程管理

章节大纲

Josh is shopping for a koozie that is tall enough to keep a tall can of his favorite beverage, ice tea, cold. A koozie is a styrofoam container into which a can of beverage slides to keep the can insulated and, thus, the beverage cold. Josh finds a variety of koozies that are the same radius , but different heights. He isn't sure how tall his ice tea can is. He does, however, know the radius and of the can. The radius of the can is 1.5 inches and the surface area is 80 square inches. What is the height of his can of ice tea?
::乔希在买一个高到足以保持他最爱的饮料、冰茶、冷的罐头的库奇。库奇是一个泡沫容器, 里面装着一瓶饮料幻灯片, 以保持罐子隔热, 从而保持饮料冷。乔希发现了各种圆形相同的库奇, 但高度不同。他不确定他的冰茶能有多高。但是, 他知道罐子的半径和罐头。罐子的半径是1.5英寸, 表面面积是80平方英寸。他冰茶罐的高度是多少?

In this concept, you will learn how to calculate the height of a cylinder given the radius and surface area.
::在此概念中, 您将学习如何计算圆柱体的高度, 取决于圆柱体的半径和表面面积。

Finding the Height of a Cylinder Given Surface Area
::寻找气团地表面积的高度

Sometimes you are given the surface area of a cylinder and need to find its height. To do this, you simply fill the information you know into the surface area formula for a cylinder. So instead of solving for $\begin{aligned} S A \end{aligned}$ , you solve for $\begin{aligned} h \end{aligned}$ , the height of the cylinder.
::有时给您一个圆柱体的表面面积, 需要找到它的高度。为此, 您只需将您知道的信息填入圆柱体的表面积公式中。因此, 不解决 SA , 而是解决h, 圆柱体的高度。

Let's look at an example.
::让我们举个例子。

The surface area of a cylinder with a radius of 3 inches is $\begin{aligned} 78 π \end{aligned}$ square inches. What is the height of the cylinder?
::圆柱体半径为3英寸的表面面积为78-平方英寸。圆柱体的高度是多少?

First, substitute the values into the formula for surface area of a cylinder and then solve for height.
::首先,将数值替换为圆柱体表面面积的公式,然后解决高度问题。

$\begin{aligned} S A & = 2 π r^{2} + 2 π r h \\ 78 π & = 2 π (3^{2}) + 2 π (3) h \\ 78 π & = 2 π (9) + 6 π h \\ 78 π & = 18 π + 6 π h \\ 78 π - 18 π & = 6 π h Subtract 18 π from both sides. \\ 60 π & = 6 π h \\ 60 π \div 6 π & = h Divide both sides by 6 π . \\ 10 i n . & = h \end{aligned}$

::SA=2r2+2rh782(32)+2(3)h782(9)+6h7818186h7818661818618181861818618×186066610英寸

The answer is 10 inches.
::答案是10英寸

You can check your work by putting the height into the formula and solving for surface area.
::您可以通过将高度放入公式和解决表面区域来检查您的工作。

$\begin{aligned} S A & = 2 π (3^{2}) + 2 π (3) (10) \\ S A & = 2 π (9) + 2 π (30) \\ S A & = 18 π + 60 π \\ S A & = 78 π \end{aligned}$

::SA=2(32)+2(3)(3)(10)SA=2(9)+2(30)SA=1860SA=78

This is the surface area given in the problem, so the answer is correct.
::这是问题中给出的表面积, 因此答案是正确的。

Let’s look at another example.
::让我们再看看另一个例子。

Candice is going to use decoupage to decorate the outside of a cylindrical canister. The canister has a radius of 3 inches and has a surface area of $\begin{aligned} 207.24 i n^{2} \end{aligned}$ . What is the height of the canister?
::Candice将用脱阴孔来装饰圆柱体罐体的外部。罐体半径为3英寸,表面面积为207. 24英寸。罐体的高度是多少?

To figure this out, first convert the surface area value into a value times pi . This is done by dividing the surface area of 207.25 square inches by the value of pi, 3.14:
::要了解这一点,首先将表面积值转换成数值乘以 pi。这样做的方法是将207. 25平方英寸的表面积除以pi的值, 3. 14:

$\begin{aligned} \frac{207.25}{3.14} = 66 (r o u n d e d) \end{aligned}$

::207.253.14=66(四舍五入)

Then, plug in the values for radius and surface area, replacing the surface area value of 207.25 with $\begin{aligned} 66 π \end{aligned}$ :
::然后插入半径和表面面积的值,将207.25的表面面积值改为66°C:

$\begin{aligned} S A & = 2 π r^{2} + 2 π r h \\ 66 π & = 2 π (3^{2}) + 2 π (3) h \\ 66 π & = 2 π (9) + 6 π h \\ 66 π & = 18 π + 6 π h \\ 66 π - 18 π & = 6 π h Subtract 18 π from both sides. \\ 66 π & = 6 π h \\ 66 π \div 6 π & = h Divide both sides by 6 π . \\ 8 i n . & = h \end{aligned}$

::SA=2r2+2rh662(32)+2(3)h662(9)+66618666618661818181861818186186186668英寸

The answer is the height of the cylinder is $\begin{aligned} 8 i n \end{aligned}$ .
::答案是圆瓶的高度是8英寸

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Josh and his koozies.
::早些时候,你得到一个问题关于乔什和他的koozies。

What should the height of the koozie be to make sure that his can of tea fits completely into it if the surface area of the can is 80 square inches and the radius is 1.5 inches?
::如果罐子的表面面积是80平方英寸,半径是1.5英寸,那么如何确保他的茶罐完全适合它?

First, calculate the surface area as multiplied by pi so we can plug the surface area and radius values into the surface area formula for a cylinder:
::首先,计算表面积乘以 pi ,这样我们可以将表面积和半径值插入气瓶的表面积公式:

$\begin{aligned} \frac{80 {i n}^{2}}{3.14} = 25.5 \end{aligned}$

$\begin{aligned} S A & = 2 π r^{2} + 2 π r h \\ 25.5 π & = 2 π ({1.5}^{2}) + 2 π (1.5) h \end{aligned}$
A 48 π = 2 π r 2 + 2 π r h = 2 π ( 3 2 ) + 2 π ( 3 ) h
$\begin{aligned} \end{aligned}$

::80,23.14=25.5SA=2r2+2rh25.5}2(1.52)+2(1.5)hA482r2+2rh=2(32)+2(3)h

Next, complete the multiplication in the equation and subtract 4.5 pi from both sides::
::其次,完成方程中的乘法,并从两边减去4.5 pi:

$\begin{aligned} 25.5 π & = 2 π ({1.5}^{2}) + 2 π (1.5) h \\ 25.5 π & = 2 π (2.25) + 3 π h \\ 25.5 π & = 4.5 π + 3 π h \end{aligned}$

$\begin{aligned} 25 π - 4.5 π & = 3 π h \\ 20.5 π & = 3 π h \end{aligned}$

::25.522(1.52)+22(1.5)h25.52(2.25)+3h25.5453555555555555555555555555555555555555555555555555555555555555555555555555555555555555

Then, divide both sides by $\begin{aligned} 3 π \end{aligned}$ , remembering to include the unit of measurement:
$\begin{aligned} 20.5 π & = 3 π h \\ 20.5 π \div 3 π & = h \\ 6.8 i n . & = h \end{aligned}$

::然后,将两边除以 3 , 记住要包括一个测量单位: 20.5\\ 320.5\ 36. 8英寸 = h。

The answer is the height of the ice tea can is 6.8 inches. So Josh should buy a koozie with that height to make sure the can is covered and his tea stays cold.
::答案是冰茶罐的高度是6.8英寸。所以Josh应该买个高到6.8英寸的鸡尾酒,以确保罐子被覆盖,茶水保持冷。

Example 2
::例2

Find the height of a cylinder with the following measurements: $\begin{aligned} S A = 48 π, r = 3 i n \end{aligned}$ .
::以下列测量标准查找圆柱体的高度:SA=48,r=3英寸。

First, plug the surface area and radius values into the surface area formula for a cylinder and complete the multiplication:
::首先,将表面积和半径值插入气瓶的表面积公式,完成乘法:

$\begin{aligned} S A & = 2 π r^{2} + 2 π r h \\ 48 π & = 2 π (3^{2}) + 2 π (3) h \end{aligned}$

::SA=2r2+2rh482(32)+2(3)h

$\begin{aligned} 48 π & = 2 π (9) + 6 π h \\ 48 π & = 18 π + 6 π h \end{aligned}$

::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}

Next, subtract $\begin{aligned} 18 π \end{aligned}$ from both sides:
::下一步,从双方减去18:

$\begin{aligned} 48 π & = 18 π + 6 π h \\ 48 π - 18 π & = 6 π h \\ 30 π & = 6 π h \end{aligned}$

::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}4818)}6_h48 {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}4818_h48_18_6_6_6_6_6_6_6_

Then, divide both sides by $\begin{aligned} 6 π \end{aligned}$ for the answer, remembering to include the unit of measurement:
::然后,将两边除以6,以回答问题,记住要包括一个计量单位:

$\begin{aligned} 30 π & = 6 π h \\ 30 π \div 6 π & = h \\ 5 i n . & = h \end{aligned}$
The answer is the height of the cylinder is 5 inches.
::306h3065英寸回答是气瓶的高度是5英寸

Example 3
::例3

Find the height of a cylinder with the following measurements: $\begin{aligned} S A = 60 π, r = 2 i n \end{aligned}$
::以下列测量标准查找圆筒的高度:SA=60, r=2 英寸

First, plug the surface area and radius values into the surface area formula for a cylinder and complete the multiplication:
::首先,将表面积和半径值插入气瓶的表面积公式,完成乘法:

$\begin{aligned} S A & = 2 π r^{2} + 2 π r h \\ 60 π & = 2 π (2^{2}) + 2 π (2) h \end{aligned}$

$\begin{aligned} 60 π & = 2 π (4) + 4 π h \\ 60 π & = 8 π + 4 π h \end{aligned}$

::SA=2r2+2rh602(22)+2(2)h 602(4)+4h6084h

Next, subtract $\begin{aligned} 8 π \end{aligned}$ from both sides:
::接下来,从双方减去8:

$\begin{aligned} 60 π & = 8 π + 4 π h \\ 60 π - 8 π & = 4 π h \\ 52 π & = 4 π h \end{aligned}$

::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}

Then, divide both sides by $\begin{aligned} 4 π \end{aligned}$ , remembering to include the unit of measurement:
::然后,将两边除以4,记住包括计量单位:

$\begin{aligned} 52 π & = 4 π h \\ 52 π \div 4 π & = h \\ 13 i n . & = h \end{aligned}$
The answer is the height of the cylinder is 13 inches.
::回答是气瓶的高度是13英寸

Example 4
::例4

Find the height of a cylinder with the following measurements: $\begin{aligned} S A = 80 π, r = 4 m \end{aligned}$
::以下列测量标准查找圆筒的高度:SA=80,r=4米

First, plug the surface area and radius values into the surface area formula for a cylinder and complete the mutliplication:
::首先,将表面面积和半径值插入气瓶的表面面积公式,完成变异:

$\begin{aligned} S A & = 2 π r^{2} + 2 π r h \\ 80 π & = 2 π (4^{2}) + 2 π (4) h \end{aligned}$

$\begin{aligned} 80 π & = 2 π (16) + 8 π h \\ 80 π & = 32 π + 8 π h \end{aligned}$

::SA=2r2+2rh802(42)+2(4)h 802(16)+8883288h

Then, subtract $\begin{aligned} 32 π \end{aligned}$ from both sides:
::然后,从两边减去32:

$\begin{aligned} 80 π & = 32 π + 8 π h \\ 80 π - 32 π & = 8 π h \\ 48 π & = 8 π h \end{aligned}$

::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}八八八八八八八八八九 {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}八八八八八八八九

Last, divide both sides by $\begin{aligned} 8 π \end{aligned}$ , remembering to include the unit of measurement:
::最后,将双方除以8,记住包括计量单位:

$\begin{aligned} 48 π & = 8 π h \\ 48 π \div 8 π & = h \\ 6 m . & = h \end{aligned}$

::488 -h48 -8 -h6 m.=h

The answer is the height of the cylinder is 6 meters.
::答案是圆柱体的高度是6米

Example 5
::例5

Mrs. Javitz bought a container of oatmeal in the shape of a cylinder. If the container has a radius of 5 cm and a surface area of 534 square cm., what is its height?
::Javitz女士购买了一个圆柱形燕麦容器,如果该容器半径为5厘米,表面面积为534平方厘米,那么它的高度是多少?

First, calculate the surface area as multiplied by pi so you can plug the surface area and radius values into the surface area formula for a cylinder:
::首先,计算表面积乘以 pi ,这样您就可以将表面积和半径值插入气瓶的表面积公式:

$\begin{aligned} \frac{534 {c m}^{2}}{3.14} = 170 \end{aligned}$

::534厘米23.14=170

$\begin{aligned} S A & = 2 π r^{2} + 2 π r h \\ 170 π & = 2 π (5^{2}) + 2 π (5) h \end{aligned}$

::SA=2r2+2rh1702(52)+2(5)h

A 48 π = 2 π r 2 + 2 π r h = 2 π ( 3 2 ) + 2 π ( 3 ) h
$\begin{aligned} \end{aligned}$
Next, complete the multiplication in the equation and subtract 50 pi from both sides:
::A482r2+2rh=2(32)+2(3)(hn next),完成方程中的乘法,并从两边减去50 pi:

$\begin{aligned} 170 π & = 2 π (5^{2}) + 2 π (5) h \\ 170 π & = 2 π (25) + 10 π h \\ 170 π & = 50 π + 10 π h \end{aligned}$

$\begin{aligned} 170 π - 50 π & = 10 π h \\ 120 π & = 10 π h \end{aligned}$
Then, divide both sides by $\begin{aligned} 10 π \end{aligned}$ , remembering to include the unit of measurement:
::1702(52)+2(5)h17022(25)+10h170501010h170 17050 170501010010101010h然后,将两边除以101010,记住包括计量单位:

$\begin{aligned} 120 π & = 10 π h \\ 120 π \div 10 π & = h \\ 12 c m . & = h \end{aligned}$

::12010h120_10_10_h12厘米=h

The answer is the oatmeal container has a height of 12 centimeters.
::答案是燕麦容器的高度为12厘米。

Review
::回顾

Find the height of each cylinder given the surface area and one other dimension.
::查找每个圆柱体的高度,以给与表面面积和另一个维度。

$\begin{aligned} r = 5 i n \end{aligned}$ , $\begin{aligned} S A = 376.8 i n^{2} \end{aligned}$
::r=5英寸,SA=376.8英寸2

$\begin{aligned} r = 6 i n \end{aligned}$ , $\begin{aligned} S A = 527.52 i n^{2} \end{aligned}$
::r=6英寸,SA=527.52英寸2

$\begin{aligned} r = 5.5 i n \end{aligned}$ , $\begin{aligned} S A = 336.765 i n^{2} \end{aligned}$
::r=5.5英寸,SA=336.765英寸2

$\begin{aligned} r = 4 c m \end{aligned}$ , $\begin{aligned} S A = 251.2 c m^{2} \end{aligned}$
::r=4厘米,SA=251.2厘米

$\begin{aligned} r = 5 m \end{aligned}$ , $\begin{aligned} S A = 471 m^{2} \end{aligned}$
::r=5米,SA=471平方米

$\begin{aligned} r = 2 c m \end{aligned}$ , $\begin{aligned} S A = 87.92 c m^{2} \end{aligned}$
::r=2厘米,SA=87.92厘米

$\begin{aligned} r = 2 c m \end{aligned}$ , $\begin{aligned} S A = 113.04 c m^{2} \end{aligned}$
::r=2厘米,SA=113.04厘米

$\begin{aligned} d = 4 m \end{aligned}$ , $\begin{aligned} S A = 125.6 m^{2} \end{aligned}$
::d=4米,SA=125.6平方米

$\begin{aligned} d = 8 c m \end{aligned}$ , $\begin{aligned} S A = 904.32 c m^{2} \end{aligned}$
::d=8厘米,SA=904.32厘米2

$\begin{aligned} d = 6 c m \end{aligned}$ , $\begin{aligned} S A = 395.64 c m^{2} \end{aligned}$
::d=6厘米,SA=395.64厘米

$\begin{aligned} r = 3 c m \end{aligned}$ , $\begin{aligned} S A = 226.08 c m^{2} \end{aligned}$
::r=3厘米,SA=226.08厘米

$\begin{aligned} r = 14 c m \end{aligned}$ , $\begin{aligned} S A = 2637.6 c m^{2} \end{aligned}$
::r=14厘米,SA=2637.6厘米

$\begin{aligned} r = 18 c m \end{aligned}$ , $\begin{aligned} S A = 4295.52 c m^{2} \end{aligned}$
::r=18厘米,SA=4295.52厘米

$\begin{aligned} r = 12 c m \end{aligned}$ , $\begin{aligned} S A = 1959.36 c m^{2} \end{aligned}$
::r=12厘米,SA=1959.36厘米

$\begin{aligned} r = 13 c m \end{aligned}$ , $\begin{aligned} S A = 3428.88 c m^{2} \end{aligned}$
::r=13厘米,SA=3428.88厘米

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

版权所有 © 2024 由瓜瓜科技thewaytolearn.cn设计。保留所有权利。