Assignment 14

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00:16:54 `q001. Note that this assignment has 5 questions If a(n) = a(n-1) + 2^n, with a(0) = 3, then substitute in turn the values 1, 2, 3 and 4 into the equation to obtain the values a(1), a(2), a(3) and a(4).

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RESPONSE --> a(1) = a(n-1) + 2^n a(1-1) + 2^1 a + 2

a is not a number, it's the name of a function. The expression a(n) is just like the expression f(x), where a is the function and n is its argument.

So a(1-1) = a(0).

You are given a(0) = 3. So your step

a(1-1) + 2^1 become a(0) + 2 or 3 + 2 which is 5.

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00:18:44 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 we are given 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 we are given 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 we are given a(3) = 16 we now have a(4) = 17 + 16 = 33.

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RESPONSE --> I have no idea what i was thinking. I don't understand this, i think...??? i must have interpreted it all wrong.

You are misinterpreting the meaning of a. See my preceding note and let me know if this doesn't clarify the process. Also be sure to view the CD video clips in your Class Notes, where this is explained in detail.

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00:22:32 `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)?

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RESPONSE --> I'm not sure if i understand how to do this. i believe my problem is that i'm making it harder than what it really is....i do this a lot with math

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00:23:55 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.

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RESPONSE --> i think i'm starting to understanding it.

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00:35:31 `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?

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RESPONSE --> I understand how to get the slope, but i'm just not fully understanding how to solve this problem....???

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00:37:47 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 = 6. 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.

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RESPONSE --> i got the y-values for the equations right, but not the slopes. i think i understand how you got the answers.....my brain isn't working apparently

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01:08:06 `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?

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RESPONSE --> if 4in. of a sphere weighs 3lbs. then a sphere that is 2ft. in diameter would weigh 18lbs. because there are 24in. in 2 feet and 24 / 4 = 6...6 * 3 = 18.

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01:09:42 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 = 748.

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RESPONSE --> i must have misinterpreted the problem because i didn't solve it the same. ?? i'm so confused...

The key statement is

'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.'

You treated this as if w = k * d. This is not so because when a sphere is scaled up, it doesn't grow in just one direction. It gets longer, and also higher and wider. So for example if it has twice the diameter, it has 2 * 2 * 2 = 8 times the volume.

Again, be sure you pay close attention to this idea in the class notes and the worksheets.

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01:13:03 `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.

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RESPONSE --> Through solving patterns, i guess that the second box will weigh about 66 ounces. ????

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01:14:19 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.

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RESPONSE --> i believe i'm even more confused than what i was....i understand now, after reading your answers, how you solved it, but i don't know the concept of how you chose to solve them....

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Good job. Let me know if you have questions.