#$&* course Mth 158 5/22 9 Question: `q001. If you are earning money at the rate of 8 dollars / hour and work for 4 hours, how much money do you make during this time? Answer in such a way as to explain your reasoning as fully as possible. A solution to this problem appears several lines below, but enter your own solution before you look at the given solution.YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
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Given Solution: Many students simply know, at the level of common sense, that if we divide $72 by $8 / hour we get 9 hours, so 9 hours are required. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): If you are sure your solution matches the given solution, and/or are sure you completely understand the given solution, then just type in 'OK'. Otherwise you should include a self-critique. In your self-critique you should explain in your own words how your solution differs from the given solution, and demonstrate what you did not originally understand but now understand about the problem and its solution. Note that your instructor scans your document for questions and indications that you are having difficulty, usually beginning with your self-critique. • If no self-critique is present, your instructor assumes you understand the solution to your satisfaction and do not need additional information or assistance. • If you do not fully understand the given solution, and/or if you still have questions after reading and taking notes on the given solution, you should self-critique in the manner described in the preceding paragraph. Insert your 'OK' or your self-critique, as appropriate, starting in the next line: OK ********************************************* Question: `q004. Calculate (8 + 3) * 5 and 8 + 3 * 5, indicating the order of your steps. Explain, as best you can, the reasons for the difference in your results. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: (type in your solution starting in the next line) (8+3)*5 should be completed by paying attention to the order of operations. First, you should complete 8+3, which is 11. Next, you should multiple 11 by 5 equaling 55. The second equation of 8+3 * 5 should begin by multiplying 3 by 5 equaling 15. Next, 15 should be added to 8, equaling 23. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Your Confidence Rating should be entered on the line above, after the colon at the end of the prompt. Your Confidence Rating is a number from 0 to 3, which is to indicate your level of confidence in your solution. 3 means you are at least 90% confident of your solution, or that you are confident you got at least 90% of the solution 2 means that you are more that 50% confident of your solution, or that you are confident you got at least 50% of the solution 1 means that you think you probably got at least some of the solution correct but don't think you got the whole thing 0 means that you're pretty sure you didn't get anything right)
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Given Solution: (8 + 3) * 5 and 8 + 3 * 5 To evaluate (8 + 3) * 5, you will first do the calculation in parentheses. 8 + 3 = 11, so (8 + 3) * 5 = 11 * 5 = 55. To evaluate 8 + 3 * 5 you have to decide which operation to do first, 8 + 3 or 3 * 5. You should be familiar with the order of operations, which tells you that multiplication precedes addition. The first calculation to do is therefore 3 * 5, which is equal to 15. Thus 8 + 3 * 5 = 8 + 15 = 23 The results are different because the grouping in the first expression dictates that the addition be done first. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): If you are sure your solution matches the given solution, and/or are sure you completely understand the given solution, then just type in 'OK'. Otherwise you should include a self-critique. In your self-critique you should explain in your own words how your solution differs from the given solution, and demonstrate what you did not originally understand but now understand about the problem and its solution. Note that your instructor scans your document for questions and indications that you are having difficulty, usually beginning with your self-critique. • If no self-critique is present, your instructor assumes you understand the solution to your satisfaction and do not need additional information or assistance. • If you do not fully understand the given solution, and/or if you still have questions after reading and taking notes on the given solution, you should self-critique in the manner described in the preceding paragraph. Insert your 'OK' or your self-critique, as appropriate, starting in the next line: OK ********************************************* Question: `q005. Calculate (2^4) * 3 and 2^(4 * 3), indicating the order of your steps. Explain, as best you can, the reasons for the difference in your results. Note that the symbol '^' indicates raising to a power. For example, 4^3 means 4 raised to the third power, which is the same as 4 * 4 * 4 = 64. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The first equation of (2^4) * 3 begins by figuring out the exponent, which would equal 16. Next, the 3 is multiplied into the parentheses equaling 48. The second equation of 2^(4*3) should begin by completing the equation in the parentheses. This results in 2^12. Exponents should be completed next making the final product 4096. These equations are different based on the order of operations in each question. The first one the exponents are completed first because there is not anything to do to the parentheses. The second equation requires you to find the solution to 4*3 before completing the exponent. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Self-Critique: OK ********************************************* Question: `q006. Calculate 3 * 5 - 4 * 3 ^ 2 and 3 * 5 - (4 * 3)^2 according to the standard order of operations, indicating the order of your steps. Explain, as best you can, the reasons for the difference in your results. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The first equation of 3 * 5 - 4 * 3 ^ 2 should begin by figuring out the exponent, equaling 9. Next, multiplication should be completed. The rules of the order of operations tell us to work left to right which deciding which to complete first. The result shows 15-36, with a final answer of -21. The second equation does not include working from left to right, as the first one did. The first step would be to complete the parentheses equaling 12^2. Next, the exponents should be figured out equaling 144. Next, comes the multiplication of 3*5 equaling 15, then the subtraction with a final answer of -129. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Self-Critique: OK ********************************************* Question: `q007. Let y = 2 x + 3. (Note: Liberal Arts Mathematics students are encouraged to do this problem, but are not required to do it). • Evaluate y for x = -2. What is your result? In your solution explain the steps you took to get this result. • Evaluate y for x values -1, 0, 1 and 2. Write out a copy of the table below. In your solution give the y values you obtained in your table. x y -2 -1 -1 1 0 3 1 5 2 7 Y=2(-2)+3 Complete the parentheses first by distributing in and multiplying 2 by -2, equaling -4, then add 3 equaling -1. Y=2(-1)+3 Complete the parentheses first by distributing in and multiplying 2 and -1, equaling -2, then add 3 equaling 1. Y=2(0)+3 Complete the parentheses first by distributing in and multiplying 2 and 0, equaling 0, then add 3 equaling 3. Y=2(1)+3 Complete the parentheses first by distributing in and multiplying 2 by 1 equaling 2, then add 3 equaling 5. Y=2(2)+3 Complete the parentheses first by distributing in and multiplying 2 by 2 equaling 4, then add 3 equaling 7. The graph my values show is a linear graph. The values show themselves in a line. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Self-Critique: OK ********************************************* Question: `q008. Let y = x^2 + 3. (Note: Liberal Arts Mathematics students are encouraged to do this problem, but are not required to do it). • Evaluate y for x = -2. What is your result? In your solution explain the steps you took to get this result. • Evaluate y for x values -1, 0, 1 and 2. Write out a copy of the table below. In your solution give the y values you obtained in your table. x y -2 1 -1 2 0 3 1 4 2 7 • Sketch a graph of y vs. x on a set of coordinate axes resembling the one shown below. You may of course adjust the scale of the x or the y axis to best depict the shape of your graph. -2^2+3 -1^2+3 0^2+3 1^2+3 2^2+3 -4+3=1 -1+3=2 0+3=3 1+3=4 4+3=7 The graph would be a exponential showing a semi-straight line inclining upward. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Self-Critique: In each of the equations, I multiplied the exponents incorrectly. The first one should have been -2*-2=4. I figured -2*2=-4. I did this in the first two problems, messing up my answers and my graph. ------------------------------------------------ Self-Critique Rating: 3 ********************************************* Question: `q009. Let y = 2 ^ x + 3. (Note: Liberal Arts Mathematics students are encouraged to do this problem, but are not required to do it). • Evaluate y for x = 1. What is your result? In your solution explain the steps you took to get this result. • Evaluate y for x values 2, 3 and 4. Write out a copy of the table below. In your solution give the y values you obtained in your table. x y 1 5 2 7 3 11 4 19 Sketch a graph of y vs. x on a set of coordinate axes resembling the one shown below. You may of course adjust the scale of the x or the y axis to best depict the shape of your graph. 2^1+3 2^2+3 2^3+3 2^4+3 2+3=5 4+3=7 8+3=11 16+3=19 The graph shows a linear graph. The points are all in a line. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Self-Critique: I indicated the wrong graph. My graph is poorly drawn and earlier I battled with one being exponential or linear. It ended up being linear and they looked similar so I chose linear. ------------------------------------------------ Self-Critique Rating: 3 ********************************************* Question: `q010. If you divide a certain positive number by 1, is the result greater than the original number, less than the original number or equal to the original number, or does the answer to this question depend on the original number? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The answer would always be equal to the original number. Any positive number divided by 1 will always be the original number. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Self-critique: OK ********************************************* Question: `q011. If you divide a certain positive number by a number greater than 1, is the result greater than the original number, less than the original number or equal to the original number, or does the answer to this question depend on the original number? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The answer to this question would depend on the original number. It would also depend on the number you are dividing the certain positive number by. For example, the certain positive number could be 10 and the number greater than 1 could be 2, equaling 5. Another example would be, the certain positive number being 100 and the number greater than 1 could be 10, equaling 10. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Self-Critique: I think that I made the same point that you did in the examples. In the end, it all depends on the certain numbers being used in the equation. ------------------------------------------------ Self-Critique Rating: 3 ### There is no way that the result could ever be greater than the original positive number. The result will always be less than the original number. It does not matter what the numbers are. ********************************************* Question: `q012. If you divide a certain positive number by a positive number less than 1, is the result greater than the original number, less than the original number or equal to the original number, or does the answer to this question depend on the original number? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The answer would always be larger than the original number. For example if you divided .5 by .8, it equals .625. Point 5 is our original number which the answer is more than. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Self-Critique: OK ********************************************* Question: `q013. Students often get the basic answers to nearly all, or even all these questions, correct. Your instructor has however never seen anyone who addressed all the subtleties in the given solutions in their self-critiques, and it is very common for a student to have given no self-critiques. It is very likely that there is something in the given solutions that is not expressed in your solution. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: ### There is no way that the result could ever be greater than the original positive number. The result will always be less than the original number. It does not matter what the numbers are. " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ### There is no way that the result could ever be greater than the original positive number. The result will always be less than the original number. It does not matter what the numbers are. ********************************************* Question: `q012. If you divide a certain positive number by a positive number less than 1, is the result greater than the original number, less than the original number or equal to the original number, or does the answer to this question depend on the original number? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The answer would always be larger than the original number. For example if you divided .5 by .8, it equals .625. Point 5 is our original number which the answer is more than. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Self-Critique: OK ********************************************* Question: `q013. Students often get the basic answers to nearly all, or even all these questions, correct. Your instructor has however never seen anyone who addressed all the subtleties in the given solutions in their self-critiques, and it is very common for a student to have given no self-critiques. It is very likely that there is something in the given solutions that is not expressed in your solution. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: ### There is no way that the result could ever be greater than the original positive number. The result will always be less than the original number. It does not matter what the numbers are. " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!