课程： 9.5 广场根估计

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选择节平根估计

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平根估计
Kirsten and Shelicia are making a quilted wall hanging. They are cutting out $\begin{aligned} 4^{''} \end{aligned}$ square pieces, and will sew them into a quilt that is 10 rows by 10 rows. Once the quilt pieces are sewn together, the squares will be a bit smaller because of the seams.
::柯尔斯滕和谢利西亚正在修墙。他们正在切开4平方块,然后缝成10行10行10行的缝线。一旦缝合了缝合,两方块将因缝合而小一些。

Estimate the size of the finished quilt.
::估计完成的缝合的大小。

In this concept, you will learn how to estimate squares and square roots.
::在这一概念中,你将学会如何估计平方和平方根。

Estimating Square Roots
::估计平方根

You may already know that a perfect square is a number whose square root is a whole number. You also know that every number has a square root, and most numbers are not perfect squares. If you need to find the square root of a number that is not a perfect square, you can determine which two whole numbers the root falls between.
::您可能已经知道一个完美的正方块是一个数字, 其平方根是一个整数。您也知道每个数字都有平方根, 大多数数字不是完美的正方块。如果您需要找到一个不是完美的正方块的数字的平方根, 您可以确定两个整数的根在中间。

Let’s look at an example.
::让我们举个例子。

Find $\begin{aligned} \sqrt{30} \end{aligned}$
::找找30 30

First, you know that 30 is not a perfect square because there is no whole number you can multiply by itself to equal 30.
::首先,你知道30不是一个完美的平方因为没有一个完整的数字你可以自己乘以等于30。

Next, you know the following perfect squares.
::接下来,你知道以下的完美方形。

$\begin{aligned} \begin{array}{rcl} 5 \times 5 & = & 25 \\ 6 \times 6 & = & 36 \end{array} \end{aligned}$

The answer is $\begin{aligned} \sqrt{30} \end{aligned}$ is between 5 and 6.
::答案是30在5到6之间

Another way of approximating the square root is to use a calculator. All calculators have a radical sign on them. Some of them may work differently, but the procedure below should be similar.
::平方根的另一种近似方式是使用计算器。所有计算器都有一个激进的标志。有些计算器可能工作方式不同, 但以下程序应该相似。

Find $\begin{aligned} \sqrt{30} \end{aligned}$ using a calculator.
::使用计算器找到30个

First, enter the number you would like to find the square root of, in this case, 30.
::首先,输入您想要找到的 30 的平方根。

Next, enter the radical sign.
::下一个,进入激进的标志。

Then, press the equal sign.
::然后按这个等同的牌子

The answer is 5.477225575
::答案是5.47722575

When you’re doing an estimate and writing it in an equation, it’s best to use the $\begin{aligned} \approx \end{aligned}$ symbol, which means “approximately equal to.”
::当你做一个估计并写成一个方程式时, 最好使用这个符号, 意思是“大约等于 ” 。

$\begin{aligned} \sqrt{30} \approx 5.477225575 \end{aligned}$

You can also round your answer to a decimal.
::您也可以将回答舍入小数点或小数点。

$\begin{aligned} \sqrt{30} \approx 5.5 \end{aligned}$

As you can see, 5.5 is between 5 and 6.
::如你所见,5.5在5到6之间。

You can check the answer by squaring 5.5. Remember, this is an estimate that has also been rounded.
::您可以通过对齐 5. 5 来检查答案。记住, 这是一个也已经四舍五入的估计值。

$\begin{aligned} 5.5 \times 5.5 = 30.25 \end{aligned}$

And $\begin{aligned} 30.25 \approx 30 \end{aligned}$
::还有30点25分30分

Your calculator has a $\begin{aligned} x^{2} \end{aligned}$ key to help you square decimal numbers.
::您的计算器有一个 x2 密钥来帮助您将小数数平方。

Estimate the value:
::估计值 :

$\begin{aligned} {8.2}^{2} \end{aligned}$

First, you know that 8 is the root of the perfect square 64.
::首先,你知道8是64号正方块的根

Next, enter the number 8.2
::下一步,输入数字8.2

Then hit the $\begin{aligned} x^{2} \end{aligned}$ key.
::然后按 x2 键。

The answer is 67.24.
::答案是67.24。

$\begin{aligned} {8.2}^{2} = 67.24 \end{aligned}$

Since the answer on the calculator contains only two decimal places, this is an exact answer and does not require the $\begin{aligned} \approx \end{aligned}$ sign.
::由于计算器上的答案仅包含小数点后两个位数,所以这是一个准确的答案,不需要 {} 符号。

The answer is 67.24.
::答案是67.24。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Kirsten and Shelicia, who were trying to hang a quilt on the wall.
::早些时候,你被问及Kirsten和Shelicia的问题, 他们试图在墙上挂上一件缝衣。

The girls are sewing together $\begin{aligned} 4^{''} \end{aligned}$ squares of fabric into a $\begin{aligned} 10 piece \times 10 piece \end{aligned}$ quilt but some of the length will be lost due to the seams.
::女孩们正在一起缝4平方格的织物,缝成10片×10片缝合,但部分长度将因缝合而丧失。

First, use the formula:
::首先,使用公式:

$\begin{aligned} A = s^{2} \end{aligned}$

::A=s2 阿=2

Next, fill in what you know, and perform the calculations according to the order of operations.
::接下来,填写您知道的内容,并根据操作顺序进行计算。

$\begin{aligned} \begin{array}{rcl} A & = & (10 \times 4)^{2} \\ A & = & 40^{2} \\ A & = & 1600 s q i n \end{array} \end{aligned}$

::A=( 10x4) 2A=402A=1600 平方厘米

Then, you know that 1600 is a perfect square, and the girls’ quilt will be somewhat smaller once the squares are sewn together.
::然后,你知道,1600是一个完美的广场, 女孩的缝线一旦缝合,就会小一些。

The answer is $\begin{aligned} A \approx 1600 s q i n \end{aligned}$ .
::答案是A1600平方米英寸

Example 2
::例2

Estimate $\begin{aligned} \sqrt{11} \end{aligned}$ to two decimals
::估计小数点为11到2

First, recognize that 11 is between the perfect squares of 9 and 16, so you know that the answer is somewhere between their roots of 3 and 4, respectively.
::首先,承认11是在9和16的完美方形之间, 所以你知道答案在3和4之间。

Next, use your calculator.
::下一位,用你的计算器

$\begin{aligned} \sqrt{11} \approx 3.31662479 \end{aligned}$

Then round to two decimal places.
::然后四舍五入到小数点后两位位数。

The answer is $\begin{aligned} \sqrt{11} \approx 3.32 \end{aligned}$ .
::答案是113.32。

The estimation of a number between 3 and 4 is correct.
::对3至4之间的数字的估计是正确的。

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

Estimate the value of $\begin{aligned} {4.9}^{2} \end{aligned}$ .
::估计价值为4.92。

First, you know that 4.9 is very close to 5 and the perfect square of 25.
::首先,你知道4.9非常接近5 和25的完美正方形。

Next, use your calculator.
::下一位,用你的计算器

$\begin{aligned} {4.9}^{2} = 24.01 \end{aligned}$

The answer is 24.01.
::答案是24.01。

Example 3
::例3

What is $\begin{aligned} \sqrt{99} \end{aligned}$ to two decimal places?
::小数点后99到2位是什么?

First, you know that 99 is very close to 100, the perfect square of $\begin{aligned} 10 \times 10 \end{aligned}$ .
::首先,你知道99非常接近100, 完美的方形是10×10。

Next, use your calculator.
::下一位,用你的计算器

$\begin{aligned} \sqrt{99} \approx 9.949874371 \end{aligned}$

Then round to two decimal places.
::然后四舍五入到小数点后两位位数。

The answer is $\begin{aligned} \sqrt{99} \approx 9.99 \end{aligned}$
::答案是999.99

That is very close to 10.
::接近10个了

Example 4
::例4

Brad wants to paint a checkerboard on a square table that covers 625 square cm. There are eight squares per side on the playing board and Brad is making each square $\begin{aligned} 3 c m \times 3 c m \end{aligned}$ . Brad has estimated that the checkerboard will fit. Is he correct?
::布拉德想在一张覆盖625平方厘米的方位上画一个棋盘。棋盘上每侧有8个方格, 布拉德每平方3厘米3厘米。布拉德估计这个棋盘合适。他正确吗 ?

First, visualize a checkerboard.
::首先,想象一下一个检查板。

Next, you know that $\begin{aligned} A = s^{2} \end{aligned}$
::接下来,你知道A=S2

Substitute what you know into the equation.
::以你所知道的代替方程式

$\begin{aligned} A = (3 \times 8)^{2} \end{aligned}$

::A=( 3x8) 2

Remembering the order of operations, multiply.
::记住操作的顺序, 乘以。

$\begin{aligned} A = (24)^{2} \end{aligned}$

::A=(24)2

The sides of the checkerboard will measure 24 cm each.
::检查板两侧将分别测量24厘米。

Then, square.
::然后,方块。

$\begin{aligned} A = 576 s q c m \end{aligned}$

::A=576 平方厘米

The answer is that the checkerboard will fit. Brad was correct.
::答案是检查板合适布拉德是对的

Example 5
::例5

Approximate this square root to the nearest hundredth.
::接近此平方根至最近的一百个。

$\begin{aligned} \sqrt{65} \end{aligned}$

First, recognize that 65 is not a perfect square.
::首先,认识到65不是一个完美的广场。

Next, use your calculator.
::下一位,用你的计算器

$\begin{aligned} \sqrt{65} \approx 8.062257748 \end{aligned}$

Then, round to hundredths.
::然后, 圆到一百倍。

$\begin{aligned} \sqrt{65} \approx 8.06 \end{aligned}$

The answer is 8.06.
::答案是8.06。

Review
::回顾

Find the approximate square root and round to the nearest tenth.
::查找大约平方根和圆到最近的十分之一。

$\begin{aligned} \sqrt{8} \end{aligned}$

$\begin{aligned} \sqrt{11} \end{aligned}$

$\begin{aligned} \sqrt{24} \end{aligned}$

$\begin{aligned} \sqrt{31} \end{aligned}$

$\begin{aligned} \sqrt{37} \end{aligned}$

$\begin{aligned} \sqrt{43} \end{aligned}$

$\begin{aligned} \sqrt{59} \end{aligned}$

$\begin{aligned} \sqrt{67} \end{aligned}$

$\begin{aligned} \sqrt{73} \end{aligned}$

$\begin{aligned} \sqrt{80} \end{aligned}$

$\begin{aligned} \sqrt{95} \end{aligned}$

$\begin{aligned} \sqrt{99} \end{aligned}$

$\begin{aligned} \sqrt{101} \end{aligned}$

$\begin{aligned} \sqrt{150} \end{aligned}$

$\begin{aligned} \sqrt{136} \end{aligned}$

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

常规

平根估计

Estimating Square Roots ::估计平方根

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

Estimating Square Roots
::估计平方根

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