#$&* course Mth163 June 30,2013 16:10 or 4pm
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Given Solution: If we substitute n = 1 into a(n) = a(n-1) + 2^n we get a(1) = a(1-1) + 2^1 or, since 1-1 = 0 and 2^1 = 2 a(1) = a(0) + 2. Since we are given a(0) = 3 we now have a(1) = 3 + 2 = 5. If we substitute n = 2 into a(n) = a(n-1) + 2^n we get a(2) = a(2-1) + 2^2 or, since 2-1 = 1 and 2^2 = 4 a(2) = a(1) + 4. Since in the previous step we found that a(1) = 5 we now have a(2) = 5 + 4 = 9. If we substitute n = 3 into a(n) = a(n-1) + 2^n we get a(3) = a(3-1) + 2^3 or, since 3-1 = 2 and 2^3 = 8 a(3) = a(2) + 8. Since in the previous step we found that a(2) = 9 we now have a(3) = 9 + 8 = 17. If we substitute n = 4 into a(n) = a(n-1) + 2^n we get a(4) = a(4-1) + 2^4 or, since 4-1 = 3 and 2^4 = 16 a(4) = a(3) + 16. Since in the previous step we found that a(3) = 17 we now have a(4) = 17 + 16 = 33. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): &&&&& We add the answer if the last number to the answer of the present number to get the next number in the pattern &&&& ------------------------------------------------ Self-critique rating:3 ********************************************* Question: `q002. If a(n) = 2 * a(n-1) + n with a(0) = 3, then what are the values of a(1), a(2), a(3) and a(4)? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: &&&& a(1)=2[a(1-1)]+1 (a(0)=3)+(a(1)=1)=4 Our first number in our arithmetic sequence is 4 a(2)=2[a(2-1)]+2=4 a(1)+a(2)=8 So the sequence so far looks like 4,8,.... a(3)=2[a(3-1)]+3=7 a(2)+a(3)=11 4,8,11... a(4)=2[a(4-1)]+4=10 a(3)+a(4)=17 4,8,11,17 &&&&& confidence rating #$&*:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: If we substitute n = 1 into a(n) = 2 * a(n-1) + n we get a(1) = 2 * a(1-1) + 1 or since 1-1 = 0 a(1) = 2 * a(0) + 1. Since we know that a(0) = 3 we have a(1) = 2 * 3 + 1 = 7. If we substitute n = 1 into a(n) = 2 * a(n-1) + n we get a(2) = 2 * a(2-1) + 2 or since 2-1 = 1 a(2) = 2 * a(1) + 2. Since we know that a(0) = 3 we have a(2) = 2 * 7 + 2 = 16. If we substitute n = 1 into a(n) = 2 * a(n-1) + n we get a(3) = 2 * a(3-1) + 3 or since 3-1 = 2 a(3) = 2 * a(2) + 3. Since we know that a(0) = 3 we have a(3) = 2 * 16 + 3 = 35. If we substitute n = 1 into a(n) = 2 * a(n-1) + n we get a(4) = 2 * a(4-1) + 4 or since 4-1 = 3 a(4) = 2 * a(3) + 4. Since we know that a(0) = 3 we have a(4) = 2 * 35 + 4 = 74. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): &&&& So we don't multiply until last thing by the 2 &&&& ------------------------------------------------ Self-critique rating:3 ********************************************* Question: `q003. What are the average slopes of the graph of y = x^2 + x - 2 between the x = 1 and x= 3 points, between the x = 3 and x = 5 points, between the x = 5 and x = 7 points, and between the x = 7 and x = 9 points? What is the pattern of this sequence of slopes? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: &&&& y=1^2+1-2 y=0 (1,0) (3,10) (10-0)/(3-1)=(10/2)=5 average slope between x=1 and x=3 (5,28) x2,y2 (28-10)/(5-3)=18/2=9 ave slope between x=3 and x=5 (7,54) (54-28)/(7-5)=16/2=8 ave slope x=5 to x=7 (9,88) (88-54)/(9-7)=34/2=17ave slope between x=7 and x=9 5,9,8,17 4,1,9 8,3 5 &&&& confidence rating #$&*:2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: At x = 1, 3, 5 , 7 and 9 we find by substituting that y = 0, 10, 28, 54 and 88. The x = 1, 3, 5, 7 and 9 points are therefore (1,0), (3,10), (5,28), (7,54) and (9,88). The run from one point to the next is always 2. The rises are respectively 10, 18, 26 and 34. The slopes are therefore slope between x = 1 and x = 3: slope = rise / run = 10 / 2 = 5. slope between x = 3 and x = 5: slope = rise / run = 18 / 2 = 9. slope between x = 5 and x = 7: slope = rise / run = 26 / 2 = 13. slope between x = 7 and x = 9: slope = rise / run = 34 / 2 = 17. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): &&& I miss wrote 26 as 16/2=8 Not 26/2=13 So the overall slope pattern increases by 4 each time by having the run increase by 2 and the rise increase by two each time &&&& ------------------------------------------------ Self-critique rating:3 ********************************************* Question: `q004. If a solid stone sphere 4 inches in diameter weighs 3 pounds, then what would be the weight of a solid stone sphere 2 feet in diameter? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: &&&& The difference between 4inch and 24inch =6 ratio diff 3lbs(6)=weight of 2ft (24inch) would stone sphere is 18lbs &&&
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Given Solution: The volume of a sphere is proportional to the cube of its diameters, and weight is directly proportional to volume so we have the proportionality w = k d^3, where w and d stand for weight and diameter and k is the proportionality constant. Substituting the known weight and diameter we get 3 = k * 4^3, where we understand that the weight is in pounds and the diameter in inches. This gives us 3 = 64 k so that k = 3 / 64. Our proportionality equation is now w = 3/64 * d^3. So when the diameter is 2 feet, we first recall that diameter must be in inches and say that d = 24, which we then substitute to obtain w = 3/64 * 24^3. A simple calculation gives us the final weight w = 648. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): &&& When I redid this I divided each side by 64 I thought we were trying to get k by itself &&& ------------------------------------------------ Self-critique rating:1 ********************************************* Question: `q005. Two boxes are each constructed of a single layer of cardboard. The first box is 12 inches by 18 inches by 24 inches and weighs 22 ounces; the second is 36 inches by 54 inches by 72 inches. Using proportionality determine the weight of the second box. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: &&&& 22=k(18^3). 22=k(5832). k=.003772. U(unknown) = k(54^3) So (.00377)*(54^3)=U (.00377)(157464) U=593.6 &&&&
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Given Solution: The ratios 36/12, 54/18 and 72/24 of the corresponding sides are all the same and all equal to 3, so the dimensions of the sides of the second box are 3 times those of the first. Since the thickness of the cardboard is the same on both boxes, only the dimensions of the rectangular sides change. The only thing that matters, therefore, is the surface area of the box. The proportionality is therefore of the form w = k x^2, where w is the weight of the box and x stands any linear dimension. It follows that w2 / w1 = (x2 / x1)^2. Since as we just saw x2 / x1 = 3, we see that w2 / w1 = 3^2 = 9. Since w1 = 22 oz, we write this as w2 / 22 oz = 9. Multiplying both sides by 22 oz we see that w2 = 22 oz * 9 = 198 oz. STUDENT QUESTION I reduced the sides of each box by dividing each side measurement by 6. Then since we were looking for weight, I used the y = kx^3 formula to calculate the weight of the second box. I used y as the weight and x as the volume (l*w*h). Obviously my calculation was way off. Why wouldn’t this work? INSTRUCTOR COMMENT Presumably the cardboard is of the same thickness for both boxes. So the amount of cardboard is determined by the surface area of the the box, not the volume. If the box was filled with cardboard as well as being constructed of cardboard, then the proportionality with volume would be appropriate and your solution would be correct. Self cr. &&&& The difference between the cubes are each 3 length diff We know that weight two divided by weight one equals linear dimension to divided by linear dimension one raised to the second power Linear dimension two divided by linear dimension two equals 3 Therefore weight two divided by weight one equals 3^2 Weight two divided by weight one (which is 22oz) equals linear dimension two divided by linear dimension one which equals three raised tithe second power which is equal to 9 So weight two equals 22oz(9)=198oz &&&& ********************************************* Question: `q006. Between t = 1 and t = 3 the function y = .5 t^2 - 5 t + 9 has average slope -3; between t = 3 and t = 5 the average slope is -1; and between t = 5 and t = 7 the average slope is 1. What do you conjecture will be the average slope between t = 7 and t = 9? What linear function do you think gives the slope as a function of t? What do you think will be the slope at t = 14? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: &&&& [-27-(-1.5)]/(9-7)=-25.5/2=-12.75 ave slope between x=7 x=9 Demand vs selling price The average slope interval between x=12 and x= 14 is (37-21)/(14-12)=16/2=8 &&& confidence rating #$&*:2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ If you understand the assignment and were able to solve the previously given problems from your worksheets, you should be able to complete most of the following problems quickly and easily. If you experience difficulty with some of these problems, you will be given notes and we will work to resolve difficulties. ********************************************* Question: `q007. A sphere of radius .5 meters has mass 1100 kg and requires 11 ounces of paint to cover its surface. What would be the mass of a sphere of the same material whose radius is 1.3 meters, and how much paint would it take to cover its surface with a coat whose thickness was the same as that used to cover the first? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: &&&& Mass of second sphere would be 7,436 kg The average rate of the linear dimensions is 2.6 11 paint in oz* ratio diff 2.6=28.6 oz of paint to cover SA of larger sphere confidence rating #$&*:2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ------------------------------------------------ Self-critique rating:NA """ Self-critique (if necessary): ------------------------------------------------ Self-critique rating: """ Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!