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  • 用线性图表解决问题

    Problem Solving with Linear Graphs
    ::用线性图表解决问题

    In this chapter, we’ve been solving problems where quantities are linearly related to each other. In this section, we’ll look at a few examples of linear relationships that occur in real-world problems, and see how we can solve them using graphs. Remember back to our Problem Solving Plan:
    ::在本章中,我们一直在解决数量在线性上相互联系的问题。 在本节中,我们将研究在现实世界问题中出现的线性关系的几个例子,看看我们如何用图表来解决这些问题。 记得我们的问题解决计划:

    1. Understand the Problem
      ::理解问题
    2. Devise a Plan—Translate
      ::制定计划——翻译
    3. Carry Out the Plan—Solve
      ::执行计划-预备计划
    4. Look—Check and Interpret
      ::Look- 检查和解释

    Following the Problem Solving Plan 
    ::遵循解决问题计划

    1. A cell phone company is offering its costumers the following deal: You can buy a new cell phone for $60 and pay a monthly flat rate of $40 per month for unlimited calls. How much money will this deal cost you after 9 months?
    ::1. 一家手机公司向服装师提供以下交易:你可以以60美元购买一部新手机,并每月支付每月40美元的统一费率,以支付无限制电话费,在9个月后,这笔交易要花多少钱?

    Let’s follow the problem solving plan.
    ::让我们追踪解决问题的计划。

    Step 1: The phone costs $60; the calling plan costs $40 per month.
    ::第1步:电话费60美元;呼叫计划费每月40美元。

    Let x = number of months.
    ::让 x = 月数 。

    Let y = total cost.
    ::让y=总成本。

    Step 2: We can solve this problem by making a graph that shows the number of months on the horizontal axis and the cost on the vertical axis .
    ::第2步:我们可以通过绘制一个图表,显示水平轴上的月数和垂直轴上的成本来解决这个问题。

    Since you pay $60 when you get the phone, the y − intercept is (0, 60).
    ::既然你收到电话要付60美元, y - interview是 0, 60 。

    You pay $40 for each month, so the cost rises by $40 for 1 month, so the is 40.
    ::你每个月支付40美元, 一个月的成本上涨40美元, 40美元。

    We can graph this line using the slope- intercept method .
    ::我们可以使用斜坡拦截法绘制这条线的图解。

    Step 3: The question was “How much will this deal cost after 9 months?” We can now read the answer from the graph. We draw a vertical line from 9 months until it meets the graph, and then draw a horizontal line until it meets the vertical axis.
    ::第3步:问题在于“9个月后这笔交易要花多少钱?” 我们现在可以从图表中读出答案。 我们从9个月中绘制一条垂直线,直到它与图表相匹配,然后绘制一条水平线,直到它与垂直轴相匹配。

    We see that after 9 months you pay approximately $420.
    ::我们看到,9个月后,你支付 大约420美元。

    Step 4: To check if this is correct, let’s think of the deal again.
    ::第四步:检查这是否正确, 让我们再考虑一下交易。

    Originally, you pay $60 and then $40 a month for 9 months.
    ::最初,你支付60美元,然后每月40美元 每月9个月。

    Phone = $ 60 Calling plan = $ 40 × 9 = $ 360 Total cost = $ 420.

    ::电话=60美元 电话呼叫计划=40x9美元 =360美元 总计费用=420美元 电话=60Calling plan=40x9=360美元 电话=60Calling plan=40x9=360$

    The answer checks out.
    ::答案检查出来。

    2. A stretched spring has a length of 12 inches when a weight of 2 lbs is attached to the spring. The same spring has a length of 18 inches when a weight of 5 lbs is attached to the spring. What is the length of the spring when no weights are attached?
    ::2. 弹簧的长度为12英寸,当弹簧的重量为2磅时;同一弹簧的长度为18英寸,当弹簧的重量为5磅时;没有重量时,弹簧的长度是多少?

    Step 1: We know: the length of the spring = 12 inches when weight = 2 lbs
    ::步骤1:我们知道:春季的长度=12英寸,重量=2磅

    the length of the spring = 18 inches when weight = 5 lbs
    ::弹簧长度=18英寸,重量=5磅

    We want: the length of the spring when weight = 0 lbs
    ::我们希望: 当重量= 0磅时弹簧长度

    Let x = the weight attached to the spring.
    ::让 x = 弹簧的重量。

    Let y = the length of the spring.
    ::让y=春天的长度。

    Step 2: We can solve this problem by making a graph that shows the weight on the horizontal axis and the length of the spring on the vertical axis.
    ::第2步:我们可以通过绘制一个图表,显示水平轴的重量和垂直轴的弹簧长度来解决这个问题。

    We have two points we can graph:
    ::我们有两点可以说明:

    When the weight is 2 lbs, the length of the spring is 12 inches. This gives point (2, 12).
    ::当重量为2磅时,弹簧长度为12英寸。给点( 2, 12)。

    When the weight is 5 lbs, the length of the spring is 18 inches. This gives point (5, 18).
    ::当重量为5磅时,弹簧长度为18英寸。给点(5,18)。

    Graphing those two points and connecting them gives us our line.
    ::绘制这两个点的分布图 并把它们连接起来 给我们提供了我们的线条

    Step 3: The question was: “What is the length of the spring when no weights are attached?
    ::第3步:问题是:“当没有加分时,春季的长度是多少?

    We can answer this question by reading the graph we just made. When there is no weight on the spring, the x − value equals zero, so we are just looking for the y − intercept of the graph. On the graph, the y − intercept appears to be approximately 8 inches.
    ::我们可以通过阅读我们刚刚绘制的图解解答这个问题。 当弹簧没有重量时, x- 值等于零, 所以我们正在寻找图形的 y- interview。 在图表中, y- interview 大约为 8 英寸 。

    Step 4: To check if this correct, let’s think of the problem again.
    ::步骤4:检查是否正确,让我们再想想问题。

    You can see that the length of the spring goes up by 6 inches when the weight is increased by 3 lbs, so the slope of the line is 6   i n c h e s 3   l b s = 2   i n c h e s / l b .
    ::你可以看到,当重量增加3磅时,弹簧的长度会上升6英寸,所以线的斜坡是6英寸3磅=2英寸/磅。

    To find the length of the spring when there is no weight attached, we can look at the spring when there are 2 lbs attached. For each pound we take off, the spring will shorten by 2 inches. If we take off 2 lbs, the spring will be shorter by 4 inches. So, the length of the spring with no weights is 12 inches - 4 inches = 8 inches.
    ::找到弹簧的长度时, 当没有附加重量时, 我们可以看到弹簧的长度, 当有2磅。 对于每磅我们起飞, 弹簧将缩短2英寸。 如果我们卸下2磅, 弹簧将缩短4英寸。 因此, 弹簧的长度是12英寸 - 4英寸=8英寸。

    The answer checks out.
    ::答案检查出来。

    3. Christine took 1 hour to read 22 pages of Harry Potter. She has 100 pages left to read in order to finish the book. How much time should she expect to spend reading in order to finish the book?
    ::3. 克里斯蒂娜花了1小时读了22页《哈利波特》,为完成这本书,她还有100页要读,为完成这本书,她期望花多少时间阅读?

    Step 1: We know - Christine takes 1 hour to read 22 pages.
    ::步骤1:我们知道 克里斯蒂娜读22页需要1小时

    We want - How much time it takes to read 100 pages.
    ::读100页要花多少时间

    Let x = the time expressed in hours.
    ::让 x = 以小时表示的时间 。

    Let y = the number of pages.
    ::让 y = 页面数 。

    Step 2: We can solve this problem by making a graph that shows the number of hours spent reading on the horizontal axis and the number of pages on the vertical axis.
    ::第2步:我们可以通过绘制一个图表,显示在水平轴上的读数和垂直轴上的页数来解决这个问题。

    We have two points we can graph:
    ::我们有两点可以说明:

    Christine takes 1 hour to read 22 pages. This gives point (1, 22).
    ::Christine 读22页需要1小时。 这给出了点( 1, 22) 。

    A second point is not given, but we know that Christine would take 0 hours to read 0 pages. This gives point (0, 0).
    ::第二点没有给出, 但我们知道Christine读0页需要0小时。 这给出了点( 0, 0) 。

    Graphing those two points and connecting them gives us our line.
    ::绘制这两个点的分布图 并把它们连接起来 给我们提供了我们的线条

    Step 3: The question was: “How much time should Christine expect to spend reading 100 pages?” We can find the answer from reading the graph - we draw a horizontal line from 100 pages until it meets the graph and then we draw the vertical until it meets the horizontal axis. We see that it takes approximately 4.5 hours to read the remaining 100 pages.
    ::第3步:问题在于:“Christine预计阅读100页需要花多少时间?” 我们可以从图表中找到答案——我们从100页上画一条水平线,直到它与图表相匹配,然后我们再画一条垂直线,直到它与水平轴相匹配。我们看到读剩下的100页需要大约4.5小时。

    Step 4: To check if this correct, let’s think of the problem again.
    ::步骤4:检查是否正确,让我们再想想问题。

    We know that Christine reads 22 pages per hour - this is the slope of the line or the rate at which she is reading. To find how many hours it takes her to read 100 pages, we divide the number of pages by the rate. In this case, 100   p a g e s 22   p a g e s / h o u r = 4.54   h o u r s . This is very close to the answer we got from reading the graph.
    ::我们知道Christine每小时读22页- 这是行的斜坡或她读的速率。 要找到她读100页需要多少小时, 我们用速率来将页数除以页数。 在这种情况下, 每小时100页22页=4.54小时。 这非常接近我们从阅读图中得到的答案 。

    The answer checks out.
    ::答案检查出来。

    Example
    ::示例示例示例示例

    Example 1
    ::例1

    Aatif wants to buy a surfboard that costs $249. He was given a birthday present of $50 and he has a summer job that pays him $6.50 per hour. To be able to buy the surfboard, how many hours does he need to work?
    ::Aatif想买一个花费249美元的冲浪板,他得到了50美元的生日礼物,他有一个暑假工作,每小时付6.50美元。 要买到冲浪板,他需要工作多少小时?

    Step 1: We know - The surfboard costs $249.
    ::第1步:我们知道——冲浪板费用为249美元。

    Aatif has $50.
    ::阿提夫有50块

    His job pays $6.50 per hour.
    ::他的工作每小时付6.50美元。

    We want - How many hours Aatif needs to work to buy the surfboard.
    ::Aatif需要工作多少小时才能买到冲浪板

    Let x = the time expressed in hours
    ::让 x = 以小时表示的时间

    Let y = Aatif’s earnings
    ::Let y ' Atif ' 的收入

    Step 2: We can solve this problem by making a graph that shows the number of hours spent working on the horizontal axis and Aatif’s earnings on the vertical axis.
    ::步骤2:我们可以通过绘制一个图表来解决这个问题,显示在水平轴上工作的时间数和Aatif在垂直轴上的收入。

    Aatif has $50 at the beginning. This is the y − intercept: (0, 50).
    ::Aatif 开头有 50 美元。 这是 y - 拦截 : (0, 50) 。

    He earns $6.50 per hour. This is the slope of the line.
    ::他每小时挣6.50美元 这是线的斜坡

    We can graph this line using the slope-intercept method. We graph the y − intercept of (0, 50), and we know that for each unit in the horizontal direction, the line rises by 6.5 units in the vertical direction. Here is the line that describes this situation.
    ::我们可以使用斜坡截取法绘制这条线的图解。 我们用 y- 截取法绘制( 0, 50) 的图解, 我们知道对于水平方向的每个单位, 直线在垂直方向上上升了6.5 个单位。 这是描述这种情况的直线 。

    Step 3: The question was: “How many hours does Aatif need to work to buy the surfboard?”
    ::第3步:问题是:“Aatif需要工作多少小时才能买到冲浪板?”

    We find the answer from reading the graph - since the surfboard costs $249, we draw a horizontal line from $249 on the vertical axis until it meets the graph and then we draw a vertical line downwards until it meets the horizontal axis. We see that it takes approximately 31 hours to earn the money.
    ::我们从图表中找到答案—— 因为冲浪板花费了249美元, 我们从垂直轴的249美元上画一条水平线, 直到它与图形相匹配, 然后我们向下画一条垂直线直到它与水平轴相匹配。 我们发现挣钱需要大约31小时。

    Step 4: To check if this correct, let’s think of the problem again.
    ::步骤4:检查是否正确,让我们再想想问题。

    We know that Aatif has $50 and needs $249 to buy the surfboard. So, he needs to earn $ 249 − $ 50 = $ 199 from his job.
    ::我们知道Aatif有50美元,购买冲浪板需要249美元。 因此,他需要从工作中挣249-50美元=199美元。

    His job pays $6.50 per hour. To find how many hours he need to work we divide: $ 199 $ 6.50 / h o u r = 30.6   h o u r s . This is very close to the answer we got from reading the graph.
    ::他的工作每小时付6.50美元,为了找出他需要工作多少小时,我们分配:199美元6.50美元/小时=30.6小时。这非常接近我们从阅读图表得到的答案。

    The answer checks out.
    ::答案检查出来。

    Review
    ::回顾

    Solve the following problems by making a graph and reading it.
    ::通过绘制图表和阅读图表来解决以下问题。

    1. A gym is offering a deal to new members. Customers can sign up by paying a registration fee of $200 and a monthly fee of $39.
      1. How much will this membership cost a member by the end of the year?
        ::到年底,这个成员将花费多少费用?
      2. The old membership rate was $49 a month with a registration fee of $100. How much more would a year’s membership cost at that rate?
        ::旧的会员比率是每月49美元,注册费为100美元。 以这一比率计算,一年的会员费用还要多多少?
      3. Bonus: For what number of months would the two membership rates be the same?
        ::得益:两个成员比率在多少个月里会相同?

      ::体育馆正在向新成员提供一笔交易。 客户可以通过支付200美元的注册费和39美元的月费来注册。 到年底,会员需要多少费用? 旧会员每月49美元,注册费为100美元。 一年的会员费用还要用多少? 博努斯:两个会员的月费是多少?
    2. A candle is burning at a linear rate. The candle measures five inches two minutes after it was lit. It measures three inches eight minutes after it was lit.
      1. What was the original length of the candle?
        ::蜡烛的最初长度是多少?
      2. How long will it take to burn down to a half-inch stub?
        ::要多久才能烧成半英吋的根?
      3. Six half-inch stubs of candle can be melted together to make a new candle measuring 2 5 6 inches (a little wax gets lost in the process). How many stubs would it take to make three candles the size of the original candle?
        ::蜡烛的6个半英寸长的根根根可以一起熔化,以制作一个新的256英寸长的蜡烛(在这一过程中少了一根蜡烛 ) 。 做3个像原蜡烛大小的蜡烛需要多少根根根?

      ::蜡烛以线性速度燃烧。 蜡烛在点燃后两分钟用5英寸。 蜡烛在点燃后用3英寸。 蜡烛最初的长度是8分钟。 蜡烛最初的长度是多少? 将蜡烛烧成半英寸的根部需要多长时间? 6根半英寸的蜡烛可以一起熔化,以制作一个256英寸的新的蜡烛(在这一过程中,一个小蜡烛丢失了 ) 。 做3根像原蜡烛大小的蜡烛需要多少根根管?
    3. A dipped candle is made by taking a wick and dipping it repeatedly in melted wax. The candle gets a little bit thicker with each added layer of wax. After it has been dipped three times, the candle is 6.5 mm thick. After it has been dipped six times, it is 11 mm thick.
      1. How thick is the wick before the wax is added?
        ::在添加蜡之前,水管有多厚?
      2. How many times does the wick need to be dipped to create a candle 2 cm thick?
        ::水管需要浸水多少次才能造出2厘米厚的蜡烛?

      ::浸过的蜡烛是用一根水管做成的, 并用熔化的蜡来浸过。 蜡烛每加一层蜡就变得稍微厚一点。 在浸过三次之后, 蜡烛是6.5毫米厚的。 在浸过六次之后, 蜡烛是11毫米厚的。 加过蜡之前, 水管有多厚? 需要浸过多少次才能造出一根2厘米厚的蜡烛?
    4. Tali is trying to find the thickness of a page of his telephone book. In order to do this, he takes a measurement and finds out that 55 pages measures 1 8   i n c h . What is the thickness of one page of the phone book?
      ::Tali试图找到电话簿的厚度。为了做到这一点,他采取了一种测量方法,发现55页的厚度等于18英寸。 电话簿一页的厚度是多少?
    5. Bobby and Petra are running a lemonade stand and they charge 45 cents for each glass of lemonade. In order to break even they must make $25.
      1. How many glasses of lemonade must they sell to break even?
        ::要卖多少杯柠檬水才能扯平?
      2. When they’ve sold $18 worth of lemonade, they realize that they only have enough lemons left to make 10 more glasses. To break even now, they’ll need to sell those last 10 glasses at a higher price. What does the new price need to be?
        ::当他们卖掉价值18美元的柠檬汁时,他们意识到他们只有足够的柠檬可以再做10杯。 即使是现在,他们也要以更高的价格卖掉最后10杯。 新的价格需要什么?

      ::鲍比和佩特拉正在经营一个柠檬水摊位,每杯柠檬水要价45美分。 为了打破这些摊位,他们必须挣25美元。他们必须卖多少杯柠檬才能打破? 当他们卖了价值18美元的柠檬水时,他们意识到他们只有足够多的柠檬才能再做10杯。 即使是现在,他们也需要以更高的价格卖掉最后10杯。 新价格需要多少?
    6. Dale is making cookies using a recipe that calls for 2.5 cups of flour for two dozen cookies. How many cups of flour does he need to make five dozen cookies?
      ::戴尔正在做饼干 食谱需要2.5杯面粉 需要20多个饼干。他需要多少杯面粉做50多个饼干?
    7. To buy a car, Jason makes a down payment of $1500 and pays $350 per month in installments.
      1. How much money has Jason paid at the end of one year?
        ::杰森在一年结束时付了多少钱?
      2. If the total cost of the car is $8500, how long will it take Jason to finish paying it off?
        ::如果汽车总成本是8500美元, 杰森要花多久才能付清?
      3. The resale value of the car decreases by $100 each month from the original purchase price. If Jason sells the car as soon as he finishes paying it off, how much will he get for it?
        ::汽车的转售价值比原购价每月减少100美元。 如果杰森一付清车费就卖掉汽车,他能得到多少钱?

      ::买一辆车,杰森支付1500美元的首期付款,并每月支付350美元的分期付款。杰森在一年结束时支付了多少钱?如果汽车总费用为8500美元,杰森要花多长时间才能付清?汽车的转售价值每月比原购价减少100美元。如果杰森一付清车费,他能得到多少钱?
    8. Anne transplants a rose seedling in her garden. She wants to track the growth of the rose so she measures its height every week. On the third week, she finds that the rose is 10 inches tall and on the eleventh week she finds that the rose is 14 inches tall. Assuming the rose grows linearly with time, what was the height of the rose when Anne planted it?
      ::安妮在她的花园里移植了玫瑰树苗。她想跟踪玫瑰的生长,所以她每星期都测量玫瑰的高度。第三周,她发现玫瑰高10英寸,第十一周,她发现玫瑰高14英寸。假设玫瑰随着时间的线性增长,当安妮种植玫瑰时,玫瑰的高度是多少?
    9. Ravi hangs from a giant spring whose length is 5 m. When his child Nimi hangs from the spring its length is 2 m. Ravi weighs 160 lbs and Nimi weighs 40 lbs. Write the equation for this problem in slope-intercept form. What should we expect the length of the spring to be when his wife Amardeep, who weighs 140 lbs, hangs from it?
      ::Ravi挂在长5米的巨型弹簧上。当他的孩子Nimi从春天起挂时,其长度为2米。Ravi体重160磅,Nimi体重40磅。用斜坡拦截的形式写下这个问题的方程式。当他的妻子Amardepe(体重140磅)被吊起来时,我们指望春天的长度是多少?
    10. Nadia is placing different weights on a spring and measuring the length of the stretched spring. She finds that for a 100 gram weight the length of the stretched spring is 20 cm and for a 300 gram weight the length of the stretched spring is 25 cm.
      1. What is the unstretched length of the spring?
        ::泉水的长度是多少?
      2. If the spring can only stretch to twice its unstretched length before it breaks, how much weight can it hold?
        ::如果泉水在断裂前只能伸展两倍于其未伸展的长度,那么它能承受多少重量呢?

      ::Nadia将不同的重量放在弹簧上,并测量延展弹簧的长度。她发现,对于100克的重量而言,伸展弹簧的长度是20厘米,对于300克的重量,伸展弹簧的长度是25厘米。未伸展的弹簧长度是多少?如果弹簧在断裂前只能伸展到两倍的未伸展的长度,那么它能承受多少重量?
    11. Andrew is a submarine commander. He decides to surface his submarine to periscope depth. It takes him 20 minutes to get from a depth of 400 feet to a depth of 50 feet.
      1. What was the submarine’s depth five minutes after it started surfacing?
        ::潜艇开始浮出水面5分钟后深度是多少?
      2. How much longer would it take at that rate to get all the way to the surface?
        ::要多久才能到达水面?

      ::安德鲁是潜艇指挥官。他决定将潜艇上浮到潜望镜深度。他需要20分钟才能从400英尺的深度升至50英尺的深度。潜艇开始浮出水面5分钟后的深度是多少?以这种速度直达水面需要多长时间?
    12. Kiersta’s phone has completely run out of battery power when she puts it on the charger. Ten minutes later, when the phone is 40% recharged, Kiersta’s friend Danielle calls and Kiersta takes the phone off the charger to talk to her. When she hangs up 45 minutes later, her phone has 10% of its charge left. Then she gets another call from her friend Kwan.
      1. How long can she spend talking to Kwan before the battery runs out again?
        ::在电池用完之前 她还能和关聊多久?
      2. If she puts the phone back on the charger afterward, how long will it take to recharge completely?
        ::如果她在充电后把电话放回充电器上,要多久才能完全充电?

      ::Kiersta的手机在充电器上安装时完全没电了。 10分钟后,当电话充电了40%时,Kiersta的朋友Danielle打电话,Kiersta把电话从充电器上拿下来和她交谈。45分钟后,她挂掉手机的充电量还剩下10 % 。 然后,她又接到她朋友Kwan的电话。 在电池用完之前,她还能和Kwan通话多久?如果她把电话放回充电器上,要多久才能完全充电?
    13. Marji is painting a 75-foot fence. She starts applying the first coat of paint at 2 PM, and by 2:10 she has painted 30 feet of the fence. At 2:15, her husband, who paints about 2 3 as fast as she does, comes to join her.
      1. How much of the fence has Marji painted when her husband joins in?
        ::玛吉丈夫加入时 墙上涂了多少漆?
      2. When will they have painted the whole fence?
        ::他们什么时候会把整个围栏都漆成漆呢?
      3. How long will it take them to apply the second coat of paint if they work together the whole time?
        ::他们要花多长时间才能涂上第二件涂料? 如果他们整个时间都一起工作?

      ::Marji正在画75英尺长的栅栏,她于下午2点开始涂第一件漆衣,到2:10时她已经涂了30英尺长的栅栏。 2:15时,她丈夫画的比她快23英尺左右,她的丈夫来到她身边。当她丈夫加入时,Marji画了多少栅栏?他们什么时候画整件栅栏?如果他们一直一起工作,要花多长时间涂第二件漆衣?

    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 。

    Texas Instruments Resources
    ::得克萨斯州工具资源

    In the CK-12 Texas Instruments Algebra I FlexBook® resource, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See .
    ::在CK-12得克萨斯州仪器代数I FlexBook资源中,有图表计算活动,旨在补充本章某些经验教训的目标。