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17:56:47 `q001. It will be very important in this course for your instructor to see and understand the process of visualization and reasoning you use when you solve problems. This exercise is designed to give you a first experience with these ideas, and your instructor a first look at your work.
Answer the following questions and explain in commonsense terms why your answer makes sense.......!!!!!!!!...................................
RESPONSE --> ok
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17:58:40 For each question draw a picture to make sense out of the situation, and include a description of the picture.
Samples Sample question and response Question: If a bundle of shingles covers 30 square feet, how many bundles are required to cover a 600 square foot roof? Response: We might draw a picture of a rectangle representing the area, dividing the rectangle into a number of smaller rectangles each representing the area covered by a single bundle. This makes it clear that we are dividing the roof area into 1-bundle areas, and makes it clear why we are going to have to divide. Reasoning this problem out in words, we can say that a single bundle would cover 30 square feet. Two bundles would cover 60 square feet. Three bundles would cover 90 square feet. We could continue in this manner until we reach 600 square feet. However, this would be cumbersome. It is more efficient to use the ideas of multiplication and division. We imagine grouping the 600 square feet into 30 square foot patches. There will be 600 / 30 patches and each will require exactly one bundle. We therefore require 600 / 30 bundles = 20 bundles. {}Your responses might not be as clear as the above, though they might be even more clear. I won't be looking for perfection, though I wouldn't object to it, but for a first effort at visualizing a situation and communicating a reasoning process. This is not something you are used to doing and it might take a few attempts before you can achieve good results, but you will get better every time you try. {}You might be unsure of what to do on a specific question. In such a case specific questions and expressions of confusion are also acceptable responses. Such a response must include your attempts to come up with a picture and reason out an explanation. For example your response might be Sample expression of confusion: I've drawn a picture of a pile of bundles and a roof but I'm not sure how to connect the two. I tried multiplying the number of bundles by the square feet of the roof but I got 18,000, and I know it won't take 18,000 bundles to cover the roof. How do you put the area covered by a bundle together with the roof area to get the number of bundles required? A poor response would be something like 'I don't know how to do #17'. This response reveals nothing of your attempt to understand the question and the situation. Nor does it ask a specific question. Incidentally, you might be tempted to quote rules or formulas about rates and velocities in answering these questions. Don't. This exercise isn't about being able to memorize rules and quote them. It is about expanding your ability to visualize, reason and communicate.......!!!!!!!!...................................
RESPONSE --> ok, sounds fun
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18:00:40 In your own words briefly summarize the instructions and the intent of this exercise.
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RESPONSE --> We are to take each problem and try to visualize the situation, and use this visualization to assist us in figuring out the solution. The intent of this exercise is to utilize our visual capabilities for reasoning out these problems. See the problem, lay it out, and then attempt it solution. It will provide you with an idea of how our mind works and approaches problems.
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18:02:22 `q001. If you earn 50 dollars in 5 hours, at what average rate are you earning money, in dollars per hour?
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RESPONSE --> I see the fifty dollars, as well as the clock. For every 5 hours, I've earned $50. In my mind, 50/5 = $10 for every hour. This didn't take too much visualization. I immediately went to the division.
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18:04:56 If you travel 300 miles in 6 hours, at what average rate are you traveling, in miles per hour?
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RESPONSE --> I envision a road, 300 miles long. There is a point A, starting point, and point B, the 300 mile mark. I divide that road, I've made mine a straight line, into hours. So at point A, I start at 12pm, then 1pm, 2pm, 3pm, 4pm, 5pm, 6pm designates the 300 mile marker and point B. I then count back, I have 6 hours along my road, so I divide the 300 miles by the 6 hours to come up with 50 mph.
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18:07:25 `q002. If a ball rolling down a grooved track travels 40 centimeters in 5 seconds, at what average rate is the ball moving, in centimeters per second?
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RESPONSE --> I envision an inclined grooved track, the ball is at the top of this track. The length of this track is 40 cm. For each second I make a groove, and so I make 5 from the start of the track to the end. I then divide the 40 cm by the 5 seconds, to come up with 8 cm. per second.
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18:10:02 The preceding three questions illustrate the concept of a rate. In each case, to find the rate we divided the change in some quantity (the number of dollars or the distance, in these examples) by the time required for the change (the number of hours or seconds, in these examples). Explain in your own words what is meant by the idea of a rate.
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RESPONSE --> Rate is the lenght of time it takes to do various activities or functions, whether it be earning money or distance traveled. We are looking at the amount of time it takes for a unit of this change to occur. We are always asking how long it takes to do something? To earn $50 dollars, or earn $1.00, to drive 50 miles, etc.
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18:11:39 `q003. If you are earning money at the average rate of 15 dollars per hour, how much do you earn in 6 hours?
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RESPONSE --> For this, I take an image of a clock, or even a time card that you manually punch. If I started my shift at 12pm, and ended it at 6pm, for a total of 6 hours. I went home, and figured out, that at my rate of $15/for one hour of work, since I worked 6 hours, I earned 6 * 15, and that is $90 for that shift.
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18:12:43 If you are traveling at an average rate of 60 miles per hour, how far do you travel in 9 hours?
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RESPONSE --> Once again I have the road. Point A is my starting point, and I travel to point B, 9 hours later. I started at 12pm, and ended at 9pm. I have 9 divisions in my road representing each hour, and for each hour, I went 60 miles. I multiply 9 * 60 = 540 miles.
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18:17:26 `q004. If a ball travels at and average rate of 13 centimeters per second, how far does it travel in 3 seconds?
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RESPONSE --> Again, I envision a track of some sort. The ball is starting at point A and ending at point B. There are 3 sections to this track, each one representing 1 second. Each section is also 13 cm in length. I have 3 of these sections, so I multiply 3 * 13 to get 39 cm. 39 cm are traveled in 3 seconds.
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18:19:27 In the preceding three exercises you turned the concept of a rate around. You were given the rate and the change in the clock time, and you calculated the change in the quantity. Explain in your own words how this increases your understanding of the concept of a rate.
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RESPONSE --> In the first series, we were trying to figure out the individual unit of change, rate per hour, miles per hour, centimeters per second. In this exercise, we had that information. We knew the average rate for whatever activity was happening, and based on this information, we needed to figure out the total of the change. So if we are travling 13 cm/s. and we travel for 3 seconds, how far did we go?
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18:21:10 `q005. How long does it take to earn 100 dollars at an average rate of 4 dollars per hour?
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RESPONSE --> So here I take the hours first. If I earn $4.00 for 1 hour,at that rate, how long will it take me to earn $100.00. I could start by simply multiplying 4x4x4x4x4...etc. until I hit $100 or close to, or, I could divide $100 by $4, and get my answer, which is 25 hours.
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18:23:20 How long does it take to travel 500 miles at an average rate of 25 miles per hour?
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RESPONSE --> Again, like previously, I envision a road. Point A to B is 500 miles. There are many sections in between A & B, each represents 1 hour and also 25 miles. Rather than drawing until I hit 500, I could divide 500 by 25 to get 20. This represents 20 sections, and since each section represented 25 miles as well as 1 hour, we can then multiply this 20 *1, to get 20 hours.
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18:25:06 `q006. How long does it take a rolling ball to travel 80 centimeters at an average rate of 16 centimeters per second?
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RESPONSE --> Like the road, the track for the ball is 80 cm in length, and has a number of sections representing 1 second, as well as 16 cm. If we divide 80 cm/16 cm, we get 5, which indicates therea are 5 16cm sections needed to cover the 80 cm distance. Since each section also represents 1 second of time, it would take 5 seconds to roll a ball 80 cm, that is traveling at 16cm/sec.
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18:27:49 In the preceding three exercises you again expanded your concept of the idea of a rate. Explain how these problems illustrate the concept of a rate.
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RESPONSE --> This section combined what we learned from the two previous. If you have the rate, and also the total product (distance, length of time) you are trying to reach you can figure out the details of the problem. I don't think I'm explaining this well at all. There is a goal, a total, I know what I'm earning/traveling per unit of change towards that end, how long will it take to achieve it?
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}yU؞} Student Name: assignment #001 001. Rates
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18:31:06 `q001. You should copy and paste these instructions to a word processor for reference. However you can always view them, as well as everything else that has appeared in this box, by clicking the 'Display Everything' button.
1. For the next question or answer, you click on 'Next Question / Answer' button above the box at top left until a question has been posed. Once a question has been posed you are to answer before you click again on this button. 2. Before clicking for an answer, type your best answer to the current question into the box to the right, then clip on the 'Enter Answer' button. 3. After entering your answer you will click on 'Next Question / Answer' to view the answer to the question. Do not tamper with the information displayed in the left-hand box. 4. If your answer was incorrect, incomplete or would otherwise require revision, you will enter a self-critique. If you learned something from the answer, you need to restate it in your own words in order to reinforce your learning. If there is something you feel you should note for future reference, you should make a note in your own words. Go to the response box (the right-hand box) and type in a self-critique and/or notes, as appropriate. Do not copy and paste anything from the left-hand box, since that information will be saved in any case. 5. If you wish to save your response to your Notes file you may choose to click on the 'Save As Notes' button rather than the 'Enter Answer' button. Doing so will save your work for your future reference. Your work will be saved in a Notes file in the c:\vhmthphy folder. The title of the Notes file will also include the name you gave when you started the program. 6. After clicking either the 'Enter Response' or the 'Save as Notes' button, click on 'Next Question / Answer' and proceed in a similar manner. In the right-hand box briefly describe your understanding of these instructions, then click 'Enter Answer'.......!!!!!!!!...................................
RESPONSE --> ok
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18:31:14 Your answer has been noted. Enter 'ok' in the Response Box and click on Enter Response, then click on Next Question/Answer for the first real question.
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RESPONSE --> ok
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18:57:39 `q002. Note that there are 10 questions in this assignment. The questions are of increasing difficulty--the first questions are fairly easy but later questions are very tricky. The main purposes of these exercises are to refine your thinking about rates, and to see how you process challenging information. Continue as far as you can until you are completely lost. Students who are prepared for the highest-level math courses might not ever get lost.
If you make $50 in 5 hr, then at what rate are you earning money?......!!!!!!!!...................................
RESPONSE --> $50/5 hours = $10/hour
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18:59:10 The rate at which you are earning money is the number of dollars per hour you are earning. You are earning money at the rate of 50 dollars / (5 hours) = 10 dollars / hour. It is very likely that you immediately came up with the $10 / hour because almosteveryone is familiar with the concept of the pay rate, the number of dollars per hour. Note carefully that the pay rate is found by dividing the quantity earned by the time required to earn it. Time rates in general are found by dividing an accumulated quantity by the time required to accumulate it.
You need to make note of anything in the given solution that you didn't understand when you solved the problem. If new ideas have been introduced in the solution, you need to note them. If you notice an error in your own thinking then you need to note that. In your own words, explain anything you didn't already understand and save your response as Notes.......!!!!!!!!...................................
RESPONSE --> I do understand this concept. This question was in the q_a_int_pbs_communication.exe program.
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18:59:54 `q003.If you make $60,000 per year then how much do you make per month?
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RESPONSE --> $60,000 for one year, or 12 months, so you divide $60,000 by 12 to get a rate of $5000/month.
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19:00:09 Most people will very quickly see that we need to divide $60,000 by 12 months, giving us 60,000 dollars / (12 months) = 5000 dollars / month. Note that again we have found a time rate, dividing the accumulated quantity by the time required to accumulate it.
You need to make note of anything in the given solution that you didn't understand when you solved the problem. If new ideas have been introduced in the solution, you need to note them. If you notice an error in your own thinking then you need to note that. In your own words, explain anything you didn't already understand and save your response as Notes.......!!!!!!!!...................................
RESPONSE --> OK
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19:01:07 `q004. Suppose that the $60,000 is made in a year by a small business. Would be more appropriate to say that the business makes $5000 per month, or that the business makes an average of $5000 per month?
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RESPONSE --> It would be more appropriate to say the business makes an average of $5000/month. Since it is a business its revenue probably varies from month to month, as opposed to a salary which would remain constant.
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19:01:31 Small businesses do not usually make the same amount of money every month. The amount made depends on the demand for the services or commodities provided by the business, and there are often seasonal fluctuations in addition to other market fluctuations. It is almost certain that a small business making $60,000 per year will make more than $5000 in some months and less than $5000 in others. Therefore it is much more appropriate to say that the business makes and average of $5000 per month.
You need to make note of anything in the given solution that you didn't understand when you solved the problem. If new ideas have been introduced in the solution, you need to note them. If you notice an error in your own thinking then you need to note that. In your own words, explain anything you didn't already understand and save your response as Notes.......!!!!!!!!...................................
RESPONSE --> ok
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19:03:03 `q005. If you travel 300 miles in 6 hours, at what average rate are you covering distance, and why do we say average rate instead of just plain rate?
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RESPONSE --> If I travel 300 miles in 6 hours, my average rate would be 300/6 or 50 mph. We say average rate b/c unless we are on cruise control, a straight highway w/out any stops, or an unemcumbered railroad track, chances are we will not maintain a constant speed of 50mph throughouth the 300 mile stretch. It can vary.
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19:03:32 The average rate is 50 miles per hour, or 50 miles / hour. This is obtained by dividing the accumulated quantity, the 300 miles, by the time required to accumulate it, obtaining ave rate = 300 miles / ( 6 hours) = 50 miles / hour. Note that the rate at which distance is covered is called speed. The car has an average speed of 50 miles/hour. We say 'average rate' in this case because it is almost certain that slight changes in pressure on the accelerator, traffic conditions and other factors ensure that the speed will sometimes be greater than 50 miles/hour and sometimes less than 50 miles/hour; the 50 miles/hour we obtain from the given information is clearly and overall average of the velocities.
You need to make note of anything in the given solution that you didn't understand when you solved the problem. If new ideas have been introduced in the solution, you need to note them. If you notice an error in your own thinking then you need to note that. In your own words, explain anything you didn't already understand and save your response as Notes.......!!!!!!!!...................................
RESPONSE --> OK
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19:07:23 `q006. If you use 60 gallons of gasoline on a 1200 mile trip, then at what average rate are you using gasoline, with respect to miles traveled?
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RESPONSE --> If we use 60 gallons for 1200 miles, then we divide 1200 by 60 to get the average miles traveled per gallon, which is 20 miles per gallon.
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19:09:38 The rate of change of one quantity with respect to another is the change in the first quantity, divided by the change in the second. As in previous examples, we found the rate at which money was made with respect to time by dividing the amount of money made by the time required to make it.
By analogy, the rate at which we use fuel with respect to miles traveled is the change in the amount of fuel divided by the number of miles traveled. In this case we use 60 gallons of fuel in 1200 miles, so the average rate it 60 gal / (1200 miles) = .05 gallons / mile. Note that this question didn't ask for miles per gallon. Miles per gallon is an appropriate and common calculation, but it measures the rate at which miles are covered with respect to the amount of fuel used. Be sure you see the difference. Note that in this problem we again have here an example of a rate, but unlike previous instances this rate is not calculated with respect to time. This rate is calculated with respect to the amount of fuel used. We divide the accumulated quantity, in this case miles, by the amount of fuel required to cover those miles. Note that again we call the result of this problem an average rate because there are always at least subtle differences in driving conditions that require the use of more fuel on some miles than on others. It's very important to understand the phrase 'with respect to'. Whether the calculation makes sense or not, it is defined by the order of the terms. In this case gallons / mile tells you how many gallons you are burning, on the average, per mile. This concept is not as familiar as miles / gallon, but except for familiarity it's technically no more difficult. You need to make note of anything in the given solution that you didn't understand when you solved the problem. If new ideas have been introduced in the solution, you need to note them. If you notice an error in your own thinking then you need to note that. In your own words, explain anything you didn't already understand and save your response as Notes.......!!!!!!!!...................................
RESPONSE --> Aha! I had a feeling I was missing something by the phrasing of the original question. I did provide the answer of miles per gallon, but do understand you were looking for the rate of fuel used per each mile. I think I may have rushed through this question and did not read it or grasph it thoroughly.
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19:13:28 `q007. The word 'average' generally connotes something like adding two quantities and dividing by 2, or adding several quantities and dividing by the number of quantities we added. Why is it that we are calculating average rates but we aren't adding anything?
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RESPONSE --> In effect we are adding, we are just not doing that step. The problems could be set up like this:
1st hour you traveled 20 miles 2nd hour you traveled 40 miles 3rd hour you traveled 10 miles 4th hour you traveled 50 miles From this information, what is our average speed? You would then need to add the # of hours, which is 4, and the number of miles traveled, which is 120 to come up with the rate. The fact that we're receiving the information as totals, if we traveled 120 miles in 4 hours, as opposed to the format above, doesn't really change how we calculate averages..................................................
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19:13:52 The word 'average' in the context of the dollars / month, miles / gallon types of questions we have been answering was used because we expect that in different months different amounts were earned, or that over different parts of the trip the gas mileage might have varied, but that if we knew all the individual quantities (e.g., the dollars earned each month, the number of gallons used with each mile) and averaged them in the usual manner, we would get the .05 gallons / mile, or the $5000 / month. In a sense we have already added up all the dollars earned in each month, or the miles traveled on each gallon, and we have obtained the total $60,000 or 1200 miles. Thus when we divide by the number of months or the number of gallons, we are in fact calculating an average rate.
You need to make note of anything in the given solution that you didn't understand when you solved the problem. If new ideas have been introduced in the solution, you need to note them. If you notice an error in your own thinking then you need to note that. In your own words, explain anything you didn't already understand and save your response as Notes.......!!!!!!!!...................................
RESPONSE --> ok
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19:31:28 `q008. In a study of how lifting strength is influenced by various ways of training, a study group was divided into 2 subgroups of equally matched individuals. The first group did 10 pushups per day for a year and the second group did 50 pushups per day for year. At the end of the year to lifting strength of the first group averaged 147 pounds, while that of the second group averaged 162 pounds. At what average rate did lifting strength increase per daily pushup?
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RESPONSE --> To find the average rate of strength increase per daily pushup, we need to take the difference of total pushups per year, and the difference of the lifting strength, and divide the lifting strength by the difference in pushups.
Group 1 did 3650 pushups (based on 365 days) Group 2 did 18250 pushups The difference is 14600 pushups The strength difference is 15#'s. We divide 15/14600 to produce .0010 Not sure I grasped this completely..................................................
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19:32:46 The second group had 15 pounds more lifting strength as a result of doing 40 more daily pushups than the first. The desired rate is therefore 15 pounds / 40 pushups = .375 pounds / pushup.
You need to make note of anything in the given solution that you didn't understand when you solved the problem. If new ideas have been introduced in the solution, you need to note them. If you notice an error in your own thinking then you need to note that. In your own words, explain anything you didn't already understand and save your response as Notes.......!!!!!!!!...................................
RESPONSE --> Ok, I also had this solution as well, but could not let go of the length of time being 1 year and thought I needed to include that. Looking back, however, since both groups did the exercises for 1 year, it would just be the daily difference.
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19:38:07 `q009. In another part of the study, participants all did 30 pushups per day, but one group did pushups with a 10-pound weight on their shoulders while the other used a 30-pound weight. At the end of the study, the first group had an average lifting strength of 171 pounds, while the second had an average lifting strength of 188 pounds. At what average rate did lifting strength increase with respect to the added shoulder weight?
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RESPONSE --> I am tempted to set this up in a similar fashion as the last problem. If the daily difference in weight is 20#, and the overall differnence in strength is 17#, we would divide the overall by the daily to produce 0.85 increase.
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19:38:29 The difference in lifting strength was 17 pounds, as a result of a 20 pound difference in added weight. The average rate at which strength increases with respect added weight would therefore be 17 lifting pounds / (20 added pounds) = .85 lifting pounds / added pound. The strength advantage was .85 lifting pounds per pound of added weight, on the average.
You need to make note of anything in the given solution that you didn't understand when you solved the problem. If new ideas have been introduced in the solution, you need to note them. If you notice an error in your own thinking then you need to note that. In your own words, explain anything you didn't already understand and save your response as Notes.......!!!!!!!!...................................
RESPONSE --> ok
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19:44:15 `q010. During a race, a runner passes the 100-meter mark 12 seconds after the start and the 200-meter mark 22 seconds after the start. At what average rate was the runner covering distance between those two positions?
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RESPONSE --> Between the 100 meter and 200 meter mark, the runner traveled 100 meters in 10 seconds. His average rate between these 2 positions is 10m/s.
Between the start and 100 meter mar, he traveled 100 meters in 12sec, averaging 8.3 m/second. Overall, he traveled 200 meters in 22seconds, averaging 9.09 m/second..................................................
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19:44:31 The runner traveled 100 meters between the two positions, and required 10 seconds to do so. The average rate at which the runner was covering distance was therefore 100 meters / (10 seconds) = 10 meters / second. Again this is an average rate; at different positions in his stride the runner would clearly be traveling at slightly different speeds.
You need to make note of anything in the given solution that you didn't understand when you solved the problem. If new ideas have been introduced in the solution, you need to note them. If you notice an error in your own thinking then you need to note that. In your own words, explain anything you didn't already understand and save your response as Notes.......!!!!!!!!...................................
RESPONSE --> ok
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19:50:26 `q011. During a race, a runner passes the 100-meter mark moving at 10 meters / second, and the 200-meter mark moving at 9 meters / second. What is your best estimate of how long it takes the runner to cover the 100 meter distance?
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RESPONSE --> I was not sure what the question was asking. Since it says ""What is your best estimate"" I am going to assume it's an estimate of the NEXT 100 meter distance.
Based on the previous 200 meters, the runner is traveling at an average speed of 9.48 m/second. First 100 meters = 10 m/sec or 10 seconds 2nd 100 meters = 9 m/sec or 11.1 seconds Combining the two = 200 meters in 21.1 seconds for a rate of 9.48 m/sec. At this rate, we divide 100 meters by 9.48 to produce 10.55 seconds for the next 100 meters..................................................
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19:54:02 At 10 meters/sec, the runner would require 10 seconds to travel 100 meters. However the runner seems to be slowing, and will therefore require more than 10 seconds to travel the 100 meters. We don't know what the runner's average speed is, we only know that it goes from 10 m/s to 9 m/s. The simplest estimate we could make would be that the average speed is the average of 10 m/s and 9 m/s, or (10 m/s + 9 m/s ) / 2 = 9.5 m/s. Taking this approximation as the average rate, the time required to travel 100 meters will be (100 meters) / (9.5 m/s) = 10.5 sec, approx.. Note that simply averaging the 10 m/s and the 9 m/s might not be the best way to approximate the average rate--for example we if we knew enough about the situation we might expect that this runner would maintain the 10 m/s for most of the remaining 100 meters, and simply tire during the last few seconds. However we were not given this information, and we don't add extraneous assumptions without good cause. So the approximation we used here is pretty close to the best we can do with the given information.
You need to make note of anything in the given solution that you didn't understand when you solved the problem. If new ideas have been introduced in the solution, you need to note them. If you notice an error in your own thinking then you need to note that. In your own words, explain anything you didn't already understand and save your response as Notes.......!!!!!!!!...................................
RESPONSE --> I did come up with the answer, but I'm still a bit confused at what 100 meters the question is referring to? I still feel it's the 3rd leg of 100 meters, approaching the 300 meter mark. Is this correct?
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19:55:22 `q012. We just averaged two quantities, adding them in dividing by 2, to find an average rate. We didn't do that before. Why we do it now?
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RESPONSE --> We did it for the last question because we did not have any other information on which to base the average on, only those 2 figures.
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19:58:10 In previous examples the quantities weren't rates. We were given the amount of change of some accumulating quantity, and the change in time or in some other quantity on which the first was dependent (e.g., dollars and months, miles and gallons). Here we are given 2 rates, 10 m/s and 9 m/s, in a situation where we need an average rate in order to answer a question. Within this context, averaging the 2 rates was an appropriate tactic.
You need to make note of anything in the given solution that you didn't understand when you solved the problem. If new ideas have been introduced in the solution, you need to note them. If you notice an error in your own thinking then you need to note that. In your own words, explain anything you didn't already understand and save your response as Notes.......!!!!!!!!...................................
RESPONSE --> I however, took the long away around this problem. Rather than averaging the two rates, I basically created a problem similar to the ones we'd been solving. This is why I feel I may not have understood something in the phrasing of the question.
In my interpretation, I understood the runner to have completed 200 meters in the 21.1 seconds, giving the rate of 9.48 m/sec rate, which was found also by adding the 10 + 9 and dividing by 2. I just want to make sure I'm not missing something by solving it that way, as opposed to your way?............................................