1.15未知维度使用公式

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  未知维度使用公式

  

  [Figure 1]

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  Andrea is out Christmas shopping with her mom and spots a sale on Christmas lights. She asks her mom if she can get some to string around the  perimeter  of her bedroom. Her mom agrees, so they look at the packages to try to figure out how many they will need. Andrea's mom can't remember the exact  dimensions  of Andrea's room, but she remembers that it is 81  square  feet and is a square. How can Andrea figure out how many feet of Christmas lights she needs?
  ::Andrea的妈妈不记得Andrea房间的确切尺寸, 但她记得那是81平方英尺的方形。 Andrea怎么能知道她需要多少英尺的圣诞灯?

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  In this concept, you will learn how to use formulas to solve for unknown dimensions when given perimeter or  area .
  ::在此概念中,您将学习如何在给定周边或区域时使用公式解决未知维度问题。

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  Finding Unknown Dimensions Using Formulas
  ::使用公式查找未知尺寸

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  Perimeter  is the  distance  around a figure.  Area  is the number of  square units  contained inside a figure. The perimeters and areas of   and squares can be found using the formulas below.
  ::周边是图周围的距离。区域是图中包含的平方单位数量。周界和方块以及方块区域可使用以下公式查找。

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  • Rectangle  Perimeter:  $\begin{array}{r}P=2l+2w\end{array}$
    ::矩形周边:P=2升+2瓦
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  • Square Perimeter:  $\begin{array}{r}P=4s\end{array}$
    ::方形周边:P=4秒
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  • Rectangle Area:  $\begin{array}{r}A=lw\end{array}$
    ::矩形区域: A = l 湿重
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  • Square Area:  $\begin{array}{r}A=s^2\end{array}$
    ::方块面积:A=s 2

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  An  equation  is a mathematical sentence that says that two mathematical expressions are equal. The perimeter and area formulas above are examples of equations. The variables in these equations are  $\begin{array}{r}P,A,l,w,\end{array}$  and  $\begin{array}{r}s\end{array}$ .
  ::等式是一个数学句, 它表示两个数学表达式是相等的。 上面的周边和区域公式是方程式的例子。 这些方程式中的变量是 P、 A、 l、 w 和 s 。

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  Sometimes you will know the area or perimeter of a rectangle or a square, but not know information about all of its sides. In this case, you can substitute the information you know into the relevant formula and  solve  for the unknown value. Remember that when you  solve  an equation you are looking for the value of the variable that makes both sides of the equation equal.
  ::有时你会知道矩形或方形的面积或周边, 但不知道其所有侧面的信息。 在这种情况下, 您可以将您所知道的信息替换为相关公式, 并解决未知值。 记住当您解析一个公式时, 您正在寻找使方程式两侧均相等的变量值 。

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  Here is an example.
  ::举一个例子。

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  The width of a rectangle is 10 feet and the perimeter is 50 feet. What is the length of the rectangle?
  ::矩形宽度为10英尺,周界宽度为50英尺。矩形的长度是多少?

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  First, figure out which formula is relevant. This question is about the perimeter of a rectangle, so you will be using the rectangle perimeter formula. 
  ::首先, 找出哪个公式相关。 这个问题涉及矩形的周边, 所以您将使用矩形周边公式 。

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  $\begin{array}{r}P=2l+2w\end{array}$
  ::P=2 l+2 w

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  Next, figure out which variables you have values for and which variable you are solving for. The problem says that the width is 10 feet, so  $\begin{array}{r}w=10\end{array}$ . The problem also says that the perimeter is 50 feet, so  $\begin{array}{r}P=50\end{array}$ . You are solving for the length.
  ::下一步, 找出您要解决的变量, 以及您要解决的变量。 问题在于宽度是 10 英尺, 所以 w = 10 。 问题在于周界是 50 英尺, 所以 P = 50 。 您正在解决长度问题 。

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  $\begin{array}{r}P=50\end{array}$
  ::P=50

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  $\begin{array}{r}l=?\end{array}$
  ::我=?

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  $\begin{array}{r}w=10\end{array}$
  ::W = 10

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  Now, substitute the values for  $\begin{array}{r}P\end{array}$  and  $\begin{array}{r}w\end{array}$  into the formula and simplify.
  ::现在,将P和W的值替换为公式并简化。

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  Next, solve the equation. Remember you are looking for the value of  $\begin{array}{r}l\end{array}$  that will make both sides of the equation equal. First, figure out the value of the  $\begin{array}{r}2l\end{array}$  in the equation. Ask yourself a question.
  ::下一步, 解答方程 。 请记住您正在寻找使方程两侧相等的 l 值。 首先, 找出方程中 2 l 值 。 请问您自己一个问题 。

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  "50 is equal to what number plus 20?"
  ::"50等于什么数字加20?"

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  Then, answer your question. 30 plus 20 is equal to 50, so  $\begin{array}{r}2l\end{array}$  must be equal to 30.
  ::那么,回答你的问题。 30加20等于50, 所以2l必须等于30。

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  Now, you have a simpler equation to solve. 
  ::现在,你有一个简单的方程 解决。

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  $\begin{array}{r}2l=30\end{array}$
  ::2 l = 30

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  Next, ask yourself another question.
  ::接下来,问你自己另一个问题。

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  "2 times what number is equal to 30?"
  ::"2乘以数等于30?"

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  Finally, answer your question. 2 times 15 is equal to 30, so  $\begin{array}{r}l\end{array}$  must be equal to 15.
  ::最后,回答你的问题。 2乘以15等于30, 所以我必须等于15。

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  The answer is that the length of the rectangle is 15 feet.
  ::答案是矩形的长度是15英尺

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  You can check your answer by substituting all three values into the rectangle perimeter formula, evaluating, and making sure both sides of the equation end up equal to each other.
  ::您可以通过将所有三个值替换为矩形周边公式, 来检查您的答案, 评估, 并确保方程的两边最终都相等于对方 。

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  The answer is correct.
  ::答案是正确的。

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  Here is another example.
  ::下面是另一个例子。

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  If the area of a square is 144 square miles, what is the length of one of the sides of the square?
  ::如果一个广场的面积是144平方英里,那么广场两侧之一的长度是多少?

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  First, figure out which formula is relevant. This question is about the area of a square, so you will be using the square area formula.
  ::首先, 找出哪个公式相关。 这个问题涉及一个正方形的区域, 所以您将使用正方形公式 。

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  $\begin{array}{r}A=s^2\end{array}$
  ::A = s 2

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  Next, figure out which variable you have a value for and which variable you are solving for. The problem says that the area is 144 square miles, so  $\begin{array}{r}A=144\end{array}$ . You are solving for the length of the side.
  ::下一步, 找出您要解决的变量是哪个变量的值, 您要解决的变量是哪个。 问题显示区域是144平方英里, 所以 A=144。 您正在解决侧面长度 。

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  Now, substitute the value for  $\begin{array}{r}A\end{array}$  into the formula.
  ::现在,将 A 的值替换为公式 。

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  Next, solve the equation. Remember you are looking for the value of  $\begin{array}{r}s\end{array}$  that will make both sides of the equation equal. Ask yourself a question.
  ::下一步, 解答方程 。 请记住您正在寻找 s 的值, 这样会让公式的两边都相等 。 请问您自己一个问题 。

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  "144 is equal to what number times itself?"
  ::“144等于多少倍本身?”

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  Then, answer your question. 12 times 12 is equal to 144, so  $\begin{array}{r}s\end{array}$  must be equal to 12.
  ::那么,回答你的问题。 12乘以12等于144, 所以必须等于12。

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  The answer is that the side length of the square is 12 miles.
  ::答案是广场的侧长是12英里

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  Now, check your answer by substituting both values into the square area formula, evaluating, and making sure both sides of the equation end up equal to each other.
  ::现在,通过将两个值都替换成平方块公式, 来检查你的答案,评估, 并确保方程式的两边最终 彼此对等。

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  The answer is correct.
  ::答案是正确的。

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were given a problem about Andrea and the Christmas lights.
  ::早些时候,你得到一个问题 关于安德里亚和圣诞节的灯光。

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  She's buying some to string around the perimeter of her room and needs to know how many feet of lights to buy. She knows her square room has an area of 81 square feet.
  ::她要买一些来绕着房间的周围 并且要知道要买多少英尺的灯光

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  First, figure out the length of each side of her room. Her room is a square and you know its area, so you will be using the square area formula.
  ::首先,找出她房间每一侧的长度。她的房间是一个广场,你知道它的面积,所以你将使用平方面积公式。

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  $\begin{array}{r}A=s^2\end{array}$
  ::A = s 2

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  Now, you know that the area of the square is 81 square feet, so  $\begin{array}{r}A=81\end{array}$ . Substitute this value for  $\begin{array}{r}A\end{array}$  into the formula.
  ::现在,你知道广场的面积是81平方英尺, 所以A=81。在公式中将A的这个值替换为A。

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  Next, ask yourself a question to help solve the equation.
  ::接下来,问自己一个问题 来帮助解答这个方程式

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  "81 is equal to what number times itself?"
  ::"81等于多少倍本身?"

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  Then, answer your question. 9 times 9 is equal to 81, so  $\begin{array}{r}s\end{array}$  must be equal to 9. Each side of Andrea's room is 9 feet.
  ::那么,回答你的问题。 9乘以9等于81, 所以必须等于9。 Andrea的房间每侧是9英尺。

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  Now, you need to figure out the perimeter of Andrea's room so Andrea will know how many feet of lights she needs. You know the room is a square with a side length of 9 feet. Substitute  $\begin{array}{r}s=9\end{array}$  into the square perimeter formula.
  ::现在,你需要弄清楚Andrea房间的周界,让Andrea知道她需要多少英尺的灯光。你知道房间是一个方形,侧长9英尺。在方形周界公式中,替代值=9。

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  The answer is that the perimeter of her room is 36 feet, so she needs 36 feet of lights.
  ::答案是她房间的周边是36英尺 所以她需要36英尺的灯光

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  Example 2
  ::例2

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  The Hegazzi family is designing a summer garden based on the model shown below. The bottom of the plot is a square and the top of the plot is a rectangle. If the total area of both plots in the garden is 139 square feet, what is the length of one side of the square plot (indicated with a question mark below)?
  ::Hegazzi家族正在根据下面的模型设计一个夏季花园。 地底是一个广场, 地顶是一个矩形。 如果花园中两块地的总面积是139平方英尺, 则方块一边的长度是多少( 下面有一个问题标记) ?

  

  [Figure 2]

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  First, use the given information to find the area of the rectangle plot. You will then use this information to help you to find the area of the square plot, which will allow you to find the side length of the square plot.
  ::首先, 使用给定的信息查找矩形图的区域。 然后您将使用此信息帮助您找到方块的区域, 这样您就可以找到方块的侧长 。

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  The rectangle has a length of 15 feet and a width of 6 feet. The formula to find the area of a rectangle is  $\begin{array}{r}A=lw\end{array}$ . Substitute the values for the length and the width of the rectangle into the formula and evaluate to find the area.
  ::矩形的长度为15英尺,宽度为6英尺。要找到矩形区域的公式是 A = l w。将矩形的长度和宽度值替换到公式中,并进行评估以找到区域。

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  Next, find the area of the square plot. You know that the total area is 139 square feet and the area of the rectangle is 90 square feet. You also know that the total area is equal to the rectangle area plus the square area. Substitute the values that you know into this relationship.
  ::接下来, 找到方块的面积。 您知道总面积是139平方英尺, 矩形面积是90平方英尺。 您也知道总面积相当于矩形面积加上方块。 将您知道的值作为此关系中的替代值 。

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  Now, solve the equation by asking yourself a question.
  ::现在,通过问你自己一个问题来解答这个方程式

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  "139 is equal to 90 plus what number?"
  ::"139等于90加什么数字?"

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  Next, answer the question. 90 plus 10 is equal to 100. 100 plus 39 is equal to 139. So 90 plus 49 must equal 139. The area of the square plot is 49 square feet.
  ::接下来,回答问题。90+10等于100,100+39等于139 所以90+49必须等于139 方块面积为49平方英尺。

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  Now, use the area of the square plot to help you to find the side length of the square plot. Substitute the value for  $\begin{array}{r}A\end{array}$  into the square area formula.
  ::现在,使用平方块的面积来帮助您找到平方块的侧长。 将 A 的值替换为平方块公式 。

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  Next, solve the equation.
  ::下一个,解决方程式。

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  "49 is equal to what number times itself?"
  ::"49等于多少倍本身?"

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  Then, answer your question. 7 times 7 is equal to 49, so  $\begin{array}{r}s\end{array}$  must be equal to 7.
  ::那么回答你的问题 7乘以7等于49 所以必须等于7

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  The answer is that the side length of the square plot is 7 feet.
  ::答案是,平方块的侧长是7英尺。

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  Example 3
  ::例3

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  If the perimeter of a rectangle is 30 feet, with a length of 9 feet, what is the width of the rectangle?
  ::如果矩形的周界是30英尺, 长9英尺, 矩形的宽度是多少?

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  First, figure out which formula is relevant. This question is about the perimeter of a rectangle, so you will be using the rectangle perimeter formula.
  ::首先, 找出哪个公式相关。 这个问题涉及矩形的周边, 所以您将使用矩形周边公式 。

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  $\begin{array}{r}P=2l+2w\end{array}$
  ::P=2 l+2 w

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  Next, figure out which variables you have values for and which variable you are solving for. The problem says that the perimeter is 30 feet, so  $\begin{array}{r}P=30\end{array}$ . The problem also says the length is 9 feet, so  $\begin{array}{r}l=9\end{array}$ . You are solving for the width.
  ::下一步, 找出您要解决的变量, 以及您要解决的变量。 问题在于周界是 30 英尺, 所以P = 30 。 问题在于长度是 9 英尺, 所以I = 9 。 您正在解决宽度 。

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  $\begin{array}{r}P=30\end{array}$
  ::P=30

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  $\begin{array}{r}l=9\end{array}$
  ::l=9

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  $\begin{array}{r}w=?\end{array}$
  ::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}为什么?

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  Now, substitute the values for  $\begin{array}{r}P\end{array}$  and  $\begin{array}{r}l\end{array}$  into the formula and simplify.
  ::现在,将P和I的值替换为公式并简化。

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  Next, solve the equation. First, figure out the value of the  $\begin{array}{r}2w\end{array}$  in the equation. Ask yourself a question.
  ::接下来,解开方程。 首先,找出方程中 2 w 的值。 问你自己一个问题 。

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  "30 is equal to 18 plus what number?"
  ::"30等于18加什么数字?"

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  Then, answer your question. 18 plus 12 is equal to 30, so  $\begin{array}{r}2w\end{array}$  must be equal to 12.
  ::那么,回答你的问题。 18加12等于30, 所以2w必须等于12。

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  Now, you have a simpler equation to solve.
  ::现在,你有一个简单的方程 解决。

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  $\begin{array}{r}2w=12\end{array}$
  ::2 w = 12

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  Next, ask yourself another question.
  ::接下来,问你自己另一个问题。

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  "2 times what number is equal to 12?"
  ::"2倍于12的数等于12?"

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  Finally, answer your question. 2 times 6 is equal to 12, so  $\begin{array}{r}w\end{array}$  must be equal to 6.
  ::最后,回答你的问题。 2乘以6等于12, 所以W必须等于6。

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  The answer is that the width of the rectangle is 6 feet.
  ::答案是矩形宽度是6英尺

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  Now, check your answer by substituting all three values into the rectangle perimeter formula, evaluating, and making sure both sides of the equation end up equal to each other.
  ::现在,通过将所有三个值替换为矩形周边公式, 来检查你的答案,评估, 并确保方程的两边最终 彼此对等。

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  The answer is correct.
  ::答案是正确的。

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  Example 4
  ::例4

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  If the area of a square is 225 square feet, what is the length of one of its sides?
  ::如果一个平方面积为225平方英尺,其两边的长度是多少?

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  First, figure out which formula is relevant. This question is about the area of a square, so you will be using the square area formula.
  ::首先, 找出哪个公式相关。 这个问题涉及一个正方形的区域, 所以您将使用正方形公式 。

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  $\begin{array}{r}A=s^2\end{array}$
  ::A = s 2

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  Next, figure out which variable you have a value for and which variable you are solving for. The problem says that the area is 225 square feet, so  $\begin{array}{r}A=225\end{array}$ . You are solving for the length of the side.
  ::下一步, 找出您要解决的变量是哪个变量的值, 您要解决的变量是哪个。 问题显示区域是 225 平方英尺, 所以 A = 225 。 您要解决的是侧面长度 。

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  Now, substitute the value for  $\begin{array}{r}A\end{array}$  into the formula.
  ::现在,将 A 的值替换为公式 。

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  Next, solve the equation by asking yourself a question.
  ::接下来,通过问你自己一个问题来解答这个方程式

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  "225 is equal to what number times itself?"
  ::"225等于多少乘以它本身?"

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  Then, answer your question. 15 times 15 is equal to 225, so  $\begin{array}{r}s\end{array}$  must be equal to 15.
  ::那么,回答你的问题。 15乘15等于225, 所以必须等于15。

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  The answer is that the side length of the square is 15 feet.
  ::答案是广场的侧长是15英尺

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  Now, check your answer by substituting both values into the square area formula, evaluating, and making sure both sides of the equation end up equal to each other.
  ::现在,通过将两个值都替换成平方块公式, 来检查你的答案,评估, 并确保方程式的两边最终 彼此对等。

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  The answer is correct.
  ::答案是正确的。

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  Example 5
  ::例5

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  If the perimeter of a square is 24 meters, what is the length of one of the sides?
  ::如果一个广场的周界是24米,那么其中一方的长度是多少?

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  First, figure out which formula is relevant. This question is about the perimeter of a square, so you will be using the square perimeter formula.
  ::首先, 找出哪个公式相关。 这个问题涉及一个广场的周边, 所以您将使用广场的周边公式 。

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  $\begin{array}{r}P=4s\end{array}$
  ::P = 4 秒

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  Next, figure out which variable you have a value for and which variable you are solving for. The problem says that the perimeter is 24 meters, so  $\begin{array}{r}P=24\end{array}$ . You are solving for the length of the side.
  ::下一步, 找出您要解决的变量是哪个, 哪个是您要解决的变量。 问题显示周界是24米, 所以P=24。 您要解决的是侧面的长度 。

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  Now, substitute the value for  $\begin{array}{r}P\end{array}$  into the formula.
  ::现在,将 P 的值替换为公式 。

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  Next, solve the equation by asking yourself a question.
  ::接下来,通过问你自己一个问题来解答这个方程式

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  "24 is equal to 4 times what number?"
  ::"24等于什么数字的4倍?"

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  Then, answer your question. 4 times 6 is equal to 24, so  $\begin{array}{r}s\end{array}$  must be equal to 6.
  ::那么,回答你的问题。 4乘以6等于24, 所以必须等于6。

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  The answer is that the side length of the square is 6 meters.
  ::答案是广场的侧长是6米

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  Now, check your answer by substituting both values into the square perimeter formula, evaluating, and making sure both sides of the equation end up equal to each other.
  ::现在,检查你的答案 将两个值都换成 平方圆边公式, 评估,并确保 方程的两边 最终都相等于对方。

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  The answer is correct.
  ::答案是正确的。

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  Review
  ::回顾

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  Given the area, find the side length for each square.
  ::根据面积,找到每个广场的侧长。

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  1. $\begin{array}{r}A=16sq.ft.\end{array}$  
     ::A = 16 s q. f t = 16 s q. f t.
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  2. $\begin{array}{r}A=64sq.m.\end{array}$  
     ::A=64平方米/米。
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  3. $\begin{array}{r}A=100sq.miles\end{array}$  
     ::A = 100 s q. m i l e s
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  4. $\begin{array}{r}A=121sq.inches\end{array}$  
     ::A = 121 s q. i n c h e s
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  5. $\begin{array}{r}A=144sq.ft.\end{array}$  
     ::A = 144 s q. f t.

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  Given the area and length, find the width of each rectangle.
  ::根据面积和长度,找到每个矩形的宽度。

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  6. $\begin{array}{r}A=24sq.feet,\text{length}=8feet\end{array}$  
     ::A = 24 s q. f e e t, 长度 = 8 f e e e t
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  7. $\begin{array}{r}A=48sq.feet,\text{length}=12feet\end{array}$  
     ::A = 48 s q. f e e t, 长度 = 12 f e e e t
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  8. $\begin{array}{r}A=64sq.feet,\text{length}=10feet\end{array}$  
     ::A=64 sq.f e e t,长度= 10 f e e t
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  9. $\begin{array}{r}A=120sq.meters,\text{length}=40meters\end{array}$   
     ::A = 120 s q. m e t e r s, 长度 = 40 m e t e r s
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  10. $\begin{array}{r}A=130sq.feet,\text{length}=13feet\end{array}$  
      ::A = 130 s q. f e e t, 长度 = 13 f e e e t
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  11. $\begin{array}{r}A=90sq.inches,\text{length}=45inches\end{array}$  
      ::A = 90 s q. i n c h e s, 长度 = 45 i n c e s

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  Given the perimeter, find the side length of each square.
  ::根据周界 找到每个广场的侧长

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  12. $\begin{array}{r}P=48inches\end{array}$  
      ::P=48 i n c h e s
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  13. $\begin{array}{r}P=64inches\end{array}$  
      ::P=64 i n c h e s
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  14. $\begin{array}{r}P=90inches\end{array}$  
      ::P=90 i n c h e s
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  15. $\begin{array}{r}P=35feet\end{array}$  
      ::P=35 f e e e t

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  Review (Answers)
  ::回顾(答复)