Course: 7.5 涉及皮层的赤道的近似解决办法

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涉及和平的赤道的近似解决办法
Julia is training for the Pedal for Heart and Stroke charity ride. Participants can ride a 10 km, 25 km or a 50 km route. Julia has decided to ride the 25 km route. She looks at the wheels on her bicycle and wonders if the new tires will last for all the times they will have to go around to complete this ride. Julia knows the tire warranty guarantees 25 000 rotations.
::Julia正在训练Pedal的心脏和斯特罗克慈善之旅。参与者可以乘坐10公里、25公里或50公里的路线。Julia已经决定乘坐25公里的路线。她看着自行车上的轮子,想知道新轮胎是否会持续到他们完成这一旅程的所有时间。Julia知道轮胎保修保证了25 000轮转。

If her bicycle tire has a diameter of 61 cm, does the tire warranty cover enough rotations to get her to the end of the 25 km route?
::如果她的自行车轮胎直径为61厘米, 轮胎保修是否覆盖足够的旋转, 让她到达25公里路线的终点?

In this concept, you will learn to approximate solutions to equations involving irrational numbers.
::在这个概念中,你将学会如何大致解决涉及非理性数字的方程式。

Irrational Numbers
::合理数字

An irrational number is one that cannot be written as the ratio of two integers and is a non-terminating, non-repeating decimal. Most scientific calculators will display non-terminating decimals to nine places after the decimal point. If you were to use all of these decimal places when solving an equation involving irrational numbers, then your answer would be more accurate than if you rounded the decimal to the nearest hundredth. However, your answer would still be an approximation and not an accurate answer.
::不合理数字是无法以两个整数之比写成的,是非终止、不重复的十进制数字。大多数科学计算器将显示非终止小数到小数点后9位。如果您在解答涉及不合理数字的方程式时使用所有这些小数位数,那么您的回答会比四舍五入小数到最接近的百分位更准确。然而,您的回答仍然是近似而非准确答案。

It is acceptable to round irrational numbers to the nearest hundredth or thousandth when solving equations . The answer to the equation should not include an equal sign but rather the symbol $\begin{align*}\approx\end{align*}$ which means “is approximately equal to.” If the value for the variable is “approximately equal to” use this same symbol when verifying the answer using the original equation. Both sides of the equation may not be equal but their values will be very close to being equal.
::在解析方程式时,将非理性数字的圆数回圆到最接近的千分之一百或千分之一是可以接受的。方程式的答案不应该包括一个平等的符号,而应该包括“大致相等”的符号。如果变量的值“大约等于 ” , 则在使用原始方程式核查答案时使用相同的符号。方程式的两面可能并不相等,但它们的值将非常接近于等值。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Julia and her 25 km bicycle ride. Julia needs to find out how many rotations her tires will make over the 25 km ride to determine if the warranty will last until the end of the ride.
::早些时候,你遇到了关于Julia和她的25公里自行车驾驶的问题。 Julia需要知道她的轮胎在25公里的驾驶上有多少轮转,以确定保修是否持续到行程结束。

First, calculate the circumference of the bicycle tire.
::首先,计算自行车轮胎的环形。

The diameter of the tire is 61 cm. The circumference of the tire can be found using the formula :
::轮胎直径为61厘米。

$\begin{aligned} \begin{array}{rcl} C & = & π d \\ C & = & (3.14) (61) \\ C & \approx & 191.45 c m \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} C &=& \pi d\\ C &=& (3.14)(61)\\ C &\approx& 191.45 \ cm \end{array}\end{align*}$
::CdC=(3.14)(61)C=191.45厘米

Next, convert the circumference of the tire from centimeters to meters using proportions.
::接下来,用比例将轮胎的周长从厘米转换为米。

$\begin{aligned} \begin{array}{rcl} \frac{100 c m}{191.54 c m} & = & \frac{1 m}{x} \\ \frac{100 c m}{191.54 c m} & = & \frac{1 m}{x} \\ 100 x & = & 191.54 \\ \frac{\overset{1}{100} x}{100} & = & \frac{191.54}{100} \\ x & \approx & 1.9154 m \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \frac{100 \ cm}{191.54 \ cm} &=& \frac{1 \ m}{x}\\ \\ \frac{100 \ \cancel{cm}}{191.54 \ \cancel{cm}} &=& \frac{1 \ m}{x}\\ \\ 100 x &=& 191.54\\ \\ \frac{\overset{{1}}{\cancel{100}} x}{\cancel{100}} &=& \frac{191.54}{100}\\ \\ x &\approx& 1.9154 \ m \end{array}\end{align*}$
::100厘米191.54厘米=1 mx100 厘米191.54 厘米=1 mx100x=191.541001x100=191.54100x1.9154米

Next, convert the circumference of the tire from meters to kilometers using proportions.
::接下来,用比例将轮胎的周长从米转换为公里。

$\begin{aligned} \begin{array}{rcl} \frac{1000 m}{1.9154 m} & = & \frac{1 k m}{x} \\ \frac{1000 m}{1.9154 m} & = & \frac{1 k m}{x} \\ 1000 x & = & 1.9154 \\ \frac{\overset{1}{1000} x}{1000} & = & \frac{1.9154}{1000} \\ x & \approx & 0.0019154 k m \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \frac{1000 \ m}{1.9154 \ m} &=& \frac{1 \ km}{x}\\ \\ \frac{1000 \ \cancel{m}}{1.9154 \ \cancel{m}} &=& \frac{1 \ km}{x}\\ \\ 1000 x &=& 1.9154\\ \\ \frac{\overset{{1}}{\cancel{1000}} x}{\cancel{1000}} &=& \frac{1.9154}{1000}\\ \\ x &\approx& 0.0019154 \ km \end{array}\end{align*}$
::1000m1.9154m=1公里x1000m1.9154m=1公里x100000x1.91541000x100000=1.91541000x0.00154千米

Then, divide the length of the race by the circumference of the tire to determine the number of rotations. Round you answer to the nearest whole number.
::然后,将赛跑的长度除以轮胎的环绕以决定旋转次数。您可以回答最近的整数。

$\begin{aligned} \begin{array}{rcl} Number of rotations & = & \frac{25 k m}{0.0019154 k m} \\ Number of rotations & = & \frac{25 k m}{0.0019154 k m} \\ Number of rotations & = & 13, 052.104 \\ Number of rotations & \approx & 13, 052 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \text{Number of rotations} &=& \frac{25 \ km}{0.0019154 \ km}\\ \\ \text{Number of rotations} &=& \frac{25 \ \cancel{km}}{0.0019154 \ \cancel{km}}\\ \\ \text{Number of rotations} &=& 13,052.104\\ \\ \text{Number of rotations} & \approx & 13,052 \end{array}\end{align*}$
::轮调次数=25公里0.0019154公里轮调次数=25公里0.0019154公里轮调次数=13 052104公里轮调次数=13 052

The answer is approximately equal to 13,052 rotations.
::答案大约相当于13 052次轮换。

The tire warranty guarantees 25 000 rotations. Julia and her tires will make it to the end of the ride.
::轮胎保修保证有2万5千次轮转朱莉娅和她的轮胎会飞到车尾

Example 2
::例2

You can calculate the area of a circle by multiplying the radius of the circle (the distance from the center to the edge) by π (pi), which is an irrational number approximately equal to 3.14159265359. Calculations would be inconveniently complex if you used that many decimal places, so pi is commonly rounded to 3.14.
::您可以将圆的半径(从中间到边缘的距离)乘以 (pi) 来计算圆的面积,这是一个不合理的数字,大约相当于 3.14159265359。如果使用如此多的小数点,计算将会不方便地复杂, 所以 pi 通常四舍五入到 3. 14 。

Calculate the approximate area of a circle with a radius of 5 inches. Use Area = pi times radius squared, and estimate pi to two decimal places.
::计算半径为 5 英寸的圆圆的大致面积。使用区域 = pi乘以半径方形,并估计pi 到小数点后两个位数。

$\begin{aligned} A = & π \times r^{2} \end{aligned}$
$\begin{align*}A= & \pi \times r^2\\\end{align*}$
::阿 r2

First, substitute 5 in place of the $\begin{align*}r\end{align*}$ in the equation.
::首先,替代方程中的r的5。

$\begin{aligned} A = & π \times (5)^{2} \end{aligned}$
$\begin{align*} A = & \pi \times (5)^2\end{align*}$
::A (5)2

Next, simplify the radius squared. Remember that 'squared' means 'multiplied by itself'.
::接下来, 简化半径方形。请记住, “ 方形” 是指“ 本身多化 ” 。

$\begin{aligned} A = & π \times 25 \end{aligned}$
$\begin{align*}A = & \pi \times 25\end{align*}$
::A25

Then, approximate pi as 3.14 and change the equals sign to 'approximately equal'.
::然后,大约pi为3.14, 并将等值符号改为“ 约等于 ” 。

$\begin{aligned} A \approx & 3.14 \times 25 \end{aligned}$
$\begin{align*}A \approx & 3.14 \times 25\end{align*}$
::A3.14×25

Finally, multiply.
::最终,乘以乘数。

$\begin{aligned} A \approx 78.5 \end{aligned}$
$\begin{align*}A \approx 78.5\end{align*}$
::A78.5

The area of the circle is approximately 78.5 square inches.
::圆圈面积约为78.5平方英寸。

Example 3
::例3

Find the approximate volume of a pop can with a radius of 2 inches and height of 5 inches. Use the formula $\begin{align*}V = \pi \times r^2 \times h\end{align*}$ where V is the volume, r is the radius, and h is the height of the cylinder. Approximate pi as 3.14. The volume will be in inches cubed .
::查找圆柱体的大约体积, 半径为 2 英寸, 高度为 5 英寸。使用公式 V r2xh, V 是音量, r 是圆柱体的高度, h 是圆柱体的半径。 Apbearly pi 为 3. 14。音量将以英寸立方体表示。

First, substitute 2 for r , and 5 for h into the formula.
::首先,在公式中,将r改为2,h改为5。

$\begin{aligned} V = π \times 2^{2} \times 5 \end{aligned}$
$\begin{align*}V= \pi \times 2^2 \times 5\end{align*}$
::V22x5

Next, simplify the numbers.
::接下来,简化数字。

$\begin{aligned} V = & π \times 2^{2} \times 5 \\ V = & π \times 4 \times 5 \\ V = & π \times 20 \end{aligned}$
$\begin{align*}V= & \pi \times 2^2 \times 5\\ V = & \pi \times 4 \times 5\\ V = & \pi \times 20\\\end{align*}$
::V22×5V4×5V20

Then, approximate pi as 3.14, and multiply.
::然后,大约pi为3.14 乘以乘数。

$\begin{aligned} V = & π \times 20 \\ V \approx & 3.14 \times 20 \\ V \approx & 62.8 \end{aligned}$
$\begin{align*}V = & \pi \times 20\\ V \approx & 3.14 \times 20\\ V \approx & 62.8\\\end{align*}$
::V20V3.14×20V62.8

The volume of the can is approximately 62.8 inches cubed.
::罐体容量约为62.8英寸立方英寸。

Example 4
::例4

You can calculate the diagonal of a square by multiplying the length of one of the sides by $\begin{align*}\sqrt2\end{align*}$ , which can be approximated as 1.414. What is the approximate measure of the diagonal of a square with sides 6 inches long?
::您可以将方形的对角乘以两边之一的长度乘以2来计算方形的对角,该方形的长度大约为1.414。方形的对角的近度是多少,方形的对角长为6英寸?

First, write the formula for the diagonal of a square, using s for the length of a side, and d for the diagonal.
::首先,写一个方形对角的公式,用 s 表示侧面的长度, d 表示对角的长度。

$\begin{aligned} d = \sqrt{2} \times s \end{aligned}$
$\begin{align*}d= \sqrt2 \times s\end{align*}$
::d=2xs (=2xs)

Next, substitute 6 for s and 1.414 for the square root of 2.
::其次,以S6取代S,以1.414取代2平方根。

$\begin{aligned} d = & \sqrt{2} \times s \\ d = & \sqrt{2} \times 6 \\ d \approx & 1.414 \times 6 \end{aligned}$
$\begin{align*}d= & \sqrt2 \times s\\ d = & \sqrt2 \times 6\\ d \approx & 1.414 \times 6\\\end{align*}$
::d=2xsd=2x6d1.414x6

Finally, multiply.
::最终,乘以乘数。

$\begin{aligned} d \approx & 1.414 \times 6 \\ d \approx & 8.484 \end{aligned}$
$\begin{align*}d \approx & 1.414 \times 6\\ d \approx & 8.484\end{align*}$
::d1.414×6d8.484

The diagonal of the square is approximately 8.484 inches.
::广场的对角大约为8.484英寸。

Review
::回顾

Approximate the solution for each equation involving irrational numbers by rounding to the nearest hundredths place. Use 3.14 to approximate $\begin{align*}\pi\end{align*}$ and 1.41 for $\begin{align*}\sqrt{2}.\end{align*}$
::通过四舍五入到最接近的百分百位,接近每个涉及非理性数字的方程式的解决方案。使用3. 14到约和1.41来计算 2 。

$\begin{align*}a : a = 3\pi \end{align*}$
::a:a=3

$\begin{align*}x : x = 8\pi \end{align*}$
::x:x=8

$\begin{align*}b:b=9\pi \end{align*}$
::b:b=9当

$\begin{align*}c:c=12\pi \end{align*}$
::c:c=12___________________________________________________________________________________________________________________________________________________________________________________________

$\begin{align*}a:a=2\pi \end{align*}$
::a:a=2______________________________________________________________

$\begin{align*}y:y=6\pi \end{align*}$
::y:y=6

$\begin{align*}b:b=7\pi \end{align*}$
::b:b=7当

$\begin{align*}d:d=12\pi -6\end{align*}$
::d:d=126

$\begin{align*}a:a=14\pi -9\end{align*}$
::a:a=14=9

$\begin{align*}x:x=11\pi -5\end{align*}$
::x:x=11=5

$\begin{align*}\sqrt{2} +5 = x\end{align*}$
::2+5=x

$\begin{align*}8=\sqrt{2} + x\end{align*}$
::8=2+x 8=2+x

$\begin{align*}t=\pi -5.3\end{align*}$
::t5.3

$\begin{align*}\sqrt{h} = \sqrt{2} - \frac{3}{4}\end{align*}$
::h=2-34岁

Mrs. DeFazio wrote the following equation on the board: $\begin{align*}w=\sqrt{2}-{\pi}^{2}\end{align*}$ What is the value of ‘w’ in Mrs. DeFazio’s equation?
::DeFazio夫人在董事会上写了以下等式:W=22,DeFazio夫人的等式中`w ' 值是多少?

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

Section outline

General

涉及和平的赤道的近似解决办法

Irrational Numbers ::合理数字

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Review ::回顾

Review (Answers) ::回顾(答复)

Resources ::资源

Irrational Numbers
::合理数字

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Review
::回顾

Review (Answers)
::回顾(答复)

Resources
::资源