course Mth 163 }{DV[ң揞assignment #014
......!!!!!!!!...................................
10:09:21 `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).
......!!!!!!!!...................................
RESPONSE --> a(1)=a(1-1)+2^1 or a(1)=3+2=5 a(2)=a(2-1)+2^2 or a(2)=5+4=9 a(3)=a(3-1)+2^3 or a(3)=9+8=17 a(4)=a(4-1)+2^4 or a(4)=17+16=33 confidence assessment: 3
.................................................
......!!!!!!!!...................................
10:09:46 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.
......!!!!!!!!...................................
RESPONSE --> it took me a while to understand this concept but it really makes since once you think about it self critique assessment: 3
.................................................
......!!!!!!!!...................................
10:13:52 `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)?
......!!!!!!!!...................................
RESPONSE --> a(1)=2*a(1-1)+1 or a(1)=7 a(2)=2*a(2-1)+2 or a(2)=16 a(3)=2*a(3-1)+3 or a(3)=35 a(4)=2*a(4-1)+4 or a(4)=74 confidence assessment: 3
.................................................
......!!!!!!!!...................................
10:14:01 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.
......!!!!!!!!...................................
RESPONSE --> ok self critique assessment: 3
.................................................
......!!!!!!!!...................................
10:26:08 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.
......!!!!!!!!...................................
RESPONSE --> I accidentally hit the next question button and didnt feel like starting all over but this is the same answer I came up with on paper besides the 10/2= instead of 6 it should be 5...It is kindof new to me how all that worked out though. self critique assessment: 3
.................................................
......!!!!!!!!...................................
10:27:31 `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?
......!!!!!!!!...................................
RESPONSE --> I set it up as a proportion and I got that it would equal 18 pounds confidence assessment: 3
.................................................
......!!!!!!!!...................................
10:28:33 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.
......!!!!!!!!...................................
RESPONSE --> ok wow this is not how i went about it but i know now to use this next time self critique assessment: 3
.................................................
......!!!!!!!!...................................
10:33:31 `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.
......!!!!!!!!...................................
RESPONSE --> I am really lost on even finding a starting point on this one confidence assessment: 3
.................................................
......!!!!!!!!...................................
10:35:02 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.
......!!!!!!!!...................................
RESPONSE --> this makes a little more since i was just confused as to how to use the different dimensions since there was like 6 and i had no clue where to start with them but i do understand that part now self critique assessment: 3
.................................................
course Mth 163 zP}Zȓڞassignment #013
......!!!!!!!!...................................
19:28:13 `q001. Note that this assignment has 12 questions What does 2^5 mean?
......!!!!!!!!...................................
RESPONSE --> it means 2 raised to the fifth power which is literally 2*2*2*2*2 confidence assessment: 3
.................................................
......!!!!!!!!...................................
19:28:20 2^5 stands for 2 raised to the fifth power; i.e., 2^5 = 2*2*2*2*2. The result of this calculation is 2^5 = 32.
......!!!!!!!!...................................
RESPONSE --> ok self critique assessment: 3
.................................................
......!!!!!!!!...................................
19:28:51 `q002. What does 2^3 * 2^5 mean? Is the result of power of 2? If so, what power of 2 is it?
......!!!!!!!!...................................
RESPONSE --> 2^8 confidence assessment: 3
.................................................
......!!!!!!!!...................................
19:28:59 2^3 * 2^5 means (2*2*2) * (2*2*2*2*2). This is the same as 2*2*2*2*2*2*2*2, or 2^8. When we multiply this number out, we obtain 256.
......!!!!!!!!...................................
RESPONSE --> ok self critique assessment: 3
.................................................
......!!!!!!!!...................................
19:29:34 `q003. Why do we say that a^b * a^c = a^(b+c)?
......!!!!!!!!...................................
RESPONSE --> beacause when you have exponents that you are multiplying you actually add the exponents together confidence assessment: 3
.................................................
......!!!!!!!!...................................
19:29:47 We saw in the preceding example that 2^3 * 2^5 stood for a product of three 2's, multiply by a product of five 2's. We saw also that the result was identical to a product of eight 2's. This was one instance of the general rule that when we multiply to different powers of the same number, the result is that number raised to the sum of the two powers. One general way to state this rule is to let a stand for the number that is being raised to the different powers, and let b and c stand for those powers. Then we get the statement a^b * a^c = a^(b+c).
......!!!!!!!!...................................
RESPONSE --> ok self critique assessment: 3
.................................................
......!!!!!!!!...................................
19:30:51 `q004. What does (2^3)^5 mean?
......!!!!!!!!...................................
RESPONSE --> 2*2*2=8 and then 8*8*8*8*8=32768 confidence assessment: 3
.................................................
......!!!!!!!!...................................
19:31:46 Since 2^3 stands for 2*2*2, it follows that (2^3)^5 means (2^3)*(2^3)*(2^3)*(2^3)*(2^3) = (2*2*2)*(2*2*2)*(2*2*2)*(2*2*2)*(2*2*2) = 2*2*2*2*2*2*2*2*2*2*2*2*2*2*2 = 2^15.
......!!!!!!!!...................................
RESPONSE --> I would think that you would do the multiplication in the parenthesis and then have 8^5 i guess i understand this self critique assessment: 3
.................................................
......!!!!!!!!...................................
19:32:29 `q005. Why do we say that (a^b)^c = a^(b*c)?
......!!!!!!!!...................................
RESPONSE --> because in this case you multiply the two exponents together confidence assessment: 3
.................................................
......!!!!!!!!...................................
19:32:42 We saw in the last example how (2^3)^5 stands for the product of 5 quantities, each equal to the product of three 2's. We saw how this is equivalent to the product of fifteen 2's, and we saw how the fifteen was obtained by multiplying the exponents 3 and 5. In the present question a^b stands for the quantity a multiplied by itself b times. (a^b)^c stands for the quantity a^b multiplied by itself c times, which is equivalent to multiplying a by itself b * c times. Thus we say that (a^b)^c = a^(b * c).
......!!!!!!!!...................................
RESPONSE --> ok self critique assessment: 3
.................................................
......!!!!!!!!...................................
19:34:52 `q006. According to the law a^b * a^c = a*(b+c), if we multiply 2^5 by 2^-2 what power of 2 should we get?
......!!!!!!!!...................................
RESPONSE --> 8 confidence assessment: 3
.................................................
......!!!!!!!!...................................
19:35:46 To according to the law, 2^5 * 2^-2 = 2^(5 + -2) = 2^(5-2) = 2^3.
......!!!!!!!!...................................
RESPONSE --> ok this is what i did and got 8 self critique assessment: 3
.................................................
......!!!!!!!!...................................
19:36:11 `q007. Since as we saw in the preceding question 2^5 * 2^-2 = 2^3, what therefore must be the value of 2^-2?
......!!!!!!!!...................................
RESPONSE --> .25 confidence assessment: 3
.................................................
......!!!!!!!!...................................
19:36:22 One way of seeing this is to note that 2^5 = 32 and 2^3 = 8, so we have 32 * 2^-2 = 8. Dividing both sides by 32 we get 2^-2 = 8 / 32 = 1/4. We can learn something important if we keep the calculation in powers of 2. If 2^5 * 2^-2 = 2^3, then dividing both sides of the equation by 2^5 we obtain 2^-2 = 2^3/2^5, which is equal to 1/2^2. This shows us that 2^-2 = 1/2^2.
......!!!!!!!!...................................
RESPONSE --> ok self critique assessment: 3
.................................................
......!!!!!!!!...................................
19:37:27 `q008. Why must we say that 2^-n = 1 / 2^n, where n stands for any counting number?
......!!!!!!!!...................................
RESPONSE --> because when an exponent has a negative it is actually like multiplying by the recipricol confidence assessment: 3
.................................................
......!!!!!!!!...................................
19:37:48 This is because for any number m, we have 2^m * 2^-n = 2^( m + -n) = 2^(m-n), and we also have 2^m * (1 / 2^n) = 2^m / 2^n = 2^(m-n). So whether we multiply 2^m by 2^-n or by 1 / 2^n we get the same result. This shows that 2^-n and 1 / 2^n are the same.
......!!!!!!!!...................................
RESPONSE --> ok self critique assessment: 3
.................................................
......!!!!!!!!...................................
19:43:00 `q009. According to the law a^b * a^c = a*(b+c), if we multiply 2^3 by 2^-3 what power of 2 should we get? Since 2^-3 = 1 / 2^3, what number must we get when we multiply 2^3 by 2^-3?
......!!!!!!!!...................................
RESPONSE --> 2^0 confidence assessment: 3
.................................................
......!!!!!!!!...................................
19:43:14 2^3 * 2^-3 = 2^(3 + -3) = 2^(3-3) = 2^0. Since 2^-3 = 1 / 2^3 it follows that 2^3 * 2^-3 = 2^3 * ( 1 / 2^3) = 1.
......!!!!!!!!...................................
RESPONSE --> ok self critique assessment: 3
.................................................
......!!!!!!!!...................................
19:43:28 `q010. Continuing the last question, what therefore should be the value of 2^0?
......!!!!!!!!...................................
RESPONSE --> 1 confidence assessment: 3
.................................................
......!!!!!!!!...................................
19:43:39 Since 2^3 * 2^-3 = 2^0 and also 2^3 * 2^-3 = 1 we see that 2^0 must be 1.
......!!!!!!!!...................................
RESPONSE --> ok self critique assessment: 3
.................................................
......!!!!!!!!...................................
19:45:15 `q011. How do we solve the equation x^3 = 12?
......!!!!!!!!...................................
RESPONSE --> take the cube root of both sides the answer is 2.289 confidence assessment: 3
.................................................
......!!!!!!!!...................................
19:45:21 We solve the equation by taking the 1/3 power of both sides: (x^3)^(1/3) = 12^(1/3), then by the law (a^b)^c = a^(bc) we have x^(3 * 1/3) = 12^(1/3), so that x^1 = 12^(1/3) or just x = 12^(1/3), which we can easily enough evaluate with a calculator. We obtain a result of approximately x = 2.29 .
......!!!!!!!!...................................
RESPONSE --> ok self critique assessment: 3
.................................................
......!!!!!!!!...................................
19:46:50 `q012. How do we solve the equation x^(5/2) = 44?
......!!!!!!!!...................................
RESPONSE --> take the (5/2) root of each side which =4.54 confidence assessment: 3
.................................................
......!!!!!!!!...................................
19:58:14 If we take the 2/5 power of both sides we get (x^(5/2))^(2/5) = 44^(2/5) or x^(5/2 * 2/5) = 44^(2/5) or x^1 = 44^(2/5) so that x = 44^(2/5). Evaluating by calculator you should obtain approximately x = 4.54.
......!!!!!!!!...................................
RESPONSE --> ok self critique assessment: 3
.................................................