course Mth 163 ÉóëÞö™ŸÅΊv¨~ÁÛ†y€|Öºassignment #002
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16:03:11 `q001. There are 9 questions and 4 summary questions in this assignment. What is the volume of a rectangular solid whose dimensions are exactly 3 cm by 5 cm by 7 cm?
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RESPONSE --> The volume is 105cm^3. confidence assessment: 3
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16:04:38 If we orient this object so that its 3 cm dimension is its 'height', then it will be 'resting' on a rectangular base whose dimension are 5 cm by 7 cm. This base can be divided into 5 rows each consisting of 7 squares, each 1 meter by 1 meter. There will therefore be 5 * 7 = 35 such squares, showing us that the area of the base is 35 m^2. Above each of these base squares the object rises to a distance of 3 meters, forming a small rectangular tower. Each such tower can be divided into 3 cubical blocks, each having dimension 1 meter by 1 meter by 1 meter. The volume of each 1-meter cube is 1 m * 1 m * 1 m = 1 m^3, also expressed as 1 cubic meter. So each small 'tower' has volume 3 m^3. The object can be divided into 35 such 'towers'. So the total volume is 35 * 3 m^3 = 105 m^3. This construction shows us why the volume of a rectangular solid is equal to the area of the base (in this example the 35 m^2 of the base) and the altitude (in this case 3 meters). The volume of any rectangular solid is therefore V = A * h, where A is the area of the base and h the altitude. This is sometimes expressed as V = L * W * h, where L and W are the length and width of the base. However the relationship V = A * h applies to a much broader class of objects than just rectangular solids, and V = A * h is a more powerful idea than V = L * W * h. Remember both, but remember also that V = A * h is the more important.
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RESPONSE --> We have to take the 5 and multiply by the 7 to get 35m^2. Since we multiplied two meters we have to say it is m^2. We then multiply the 35 by 3 to get 105m^3. We say that it is m^3 because we multiplied m*m*m. self critique assessment: 3
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16:04:58 `q002. What is the volume of a rectangular solid whose base area is 48 square meters and whose altitude is 2 meters?
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RESPONSE --> The volume is 96m^3. confidence assessment: 3
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16:07:13 Using the idea that V = A * h we find that the volume of this solid is V = A * h = 48 m^2 * 2 m = 96 m^3. Note that m * m^2 means m * (m * m) = m * m * m = m^2.
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RESPONSE --> Since the base is 48 meters squared that means that two meters were multiplied together to get it squard. Then we multiply it by 2 to get 96m^3. That means that there were 3 meters squared m*m*m. self critique assessment: 3
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16:09:37 `q003. What is the volume of a uniform cylinder whose base area is 20 square meters and whose altitude is 40 meters?
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RESPONSE --> The volume is 50,240m^3. confidence assessment: 3
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16:11:37 V = A * h applies to uniform cylinders as well as to rectangular solids. We are given the altitude h and the base area A so we conclude that V = A * h = 20 m^2 * 40 m = 800 m^3. The relationship V = A * h applies to any solid object whose cross-sectional area A is constant. This is the case for uniform cylinders and uniform prisms.
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RESPONSE --> In this problem you just need to muliply the numbers together to get the answer. Since the 20meters is squared plus you have 40 meters then you answer is m^3. I put pi in there but there is no need for it at all and that is where I messed up at. self critique assessment: 3
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16:13:28 `q004. What is the volume of a uniform cylinder whose base has radius 5 cm and whose altitude is 30 cm?
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RESPONSE --> The volume is 2,355cm^3. confidence assessment: 3
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16:17:46 The cylinder is uniform, which means that its cross-sectional area is constant. So the relationship V = A * h applies. The cross-sectional area A is the area of a circle of radius 5 cm, so we see that A = pi r^2 = pi ( 5 cm)^2 = 25 pi cm^2. Since the altitude is 30 cm the volume is therefore V = A * h = 25 pi cm^2 * 30 cm = 750 pi cm^3. Note that the common formula for the volume of a uniform cylinder is V = pi r^2 h. However this is just an instance of the formula V = A * h, since the cross-sectional area A of the uniform cylinder is pi r^2. Rather than having to carry around the formula V = pi r^2 h, it's more efficient to remember V = A * h and to apply the well-known formula A = pi r^2 for the area of a circle.
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RESPONSE --> Instead of actually multiplying by pi we just need to square the radius of 5 to get 25 and the multiply 25 times 30 to get 750cm^3. We get cm^3 because we are using cm*cm*cm. I multiplied everything by pi=3.14. I realize that it is not necassary because this is a uniform cylinder. self critique assessment: 3
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16:22:49 `q005. Estimate the dimensions of a metal can containing food. What is its volume, as indicated by your estimates?
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RESPONSE --> The volume is about 553cm^3. confidence assessment: 2
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16:25:00 People will commonly estimate the dimensions of a can of food in centimeters or in inches, though other units of measure are possible (e.g., millimeters, feet, meters, miles, km). Different cans have different dimensions, and your estimate will depend a lot on what can you are using. A typical can might have a circular cross-section with diameter 3 inches and altitude 5 inches. This can would have volume V = A * h, where A is the area of the cross-section. The diameter of the cross-section is 3 inches so its radius will be 3/2 in.. The cross-sectional area is therefore A = pi r^2 = pi * (3/2 in)^2 = 9 pi / 4 in^2 and its volume is V = A * h = (9 pi / 4) in^2 * 5 in = 45 pi / 4 in^3. Approximating, this comes out to around 35 in^3. Another can around the same size might have diameter 8 cm and height 14 cm, giving it cross-sectional area A = pi ( 4 cm)^2 = 16 pi cm^2 and volume V = A * h = 16 pi cm^2 * 14 cm = 224 pi cm^2.
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RESPONSE --> A can can be measured in several different ways depending on what a person chooses. Someone might measuer a can in inches or someone can measure it in cm. A can can be different in diameter and in height. It just depends on how a person visualizes a can. The formula stays the same. self critique assessment: 3
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16:26:47 `q006. What is the volume of a pyramid whose base area is 50 square cm and whose altitude is 60 cm?
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RESPONSE --> The volume is 3,000cm^3. confidence assessment: 2
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16:29:40 We can't use the V = A * h idea for a pyramid because the thing doesn't have a constant cross-sectional area--from base to apex the cross-sections get smaller and smaller. It turns out that there is a way to cut up and reassemble a pyramid to show that its volume is exactly 1/3 that of a rectangular solid with base area A and altitude h. Think of putting the pyramid in a box having the same altitude as the pyramid, with the base of the pyramid just covering the bottom of the box. The apex (the point) of the pyramid will just touch the top of the box. The pyramid occupies exactly 1/3 the volume of that box. So the volume of the pyramid is V = 1/3 * A * h. The base area A is 30 cm^2 and the altitude is 60 cm so we have V = 1/3 * 50 cm^2 * 60 cm = 1000 cm^3.
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RESPONSE --> You have to figure in 1/3 of the volume because it has a base on the pyramid and the closer you get to the point thr smaller it gets. I should have divided 3000 by 1/3 but decided not too because I though it was just the base times the heighth. I now realize that you have to put the 1/3 in because of the way it get smaller the closer you go towards the point. self critique assessment: 3
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16:32:00 `q007. What is the volume of a cone whose base area is 20 square meters and whose altitude is 9 meters?
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RESPONSE --> The volume is 3,768m^3. confidence assessment: 3
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16:35:08 Just as the volume of a pyramid is 1/3 the volume of the 'box' that contains it, the volume of a cone is 1/3 the volume of the cylinder that contains it. Specifically, the cylinder that contains the cone has the base of the cone as its base and matches the altitude of the cone. So the volume of the cone is 1/3 A * h, where A is the area of the base and h is the altitude of the cone. In this case the base area and altitude are given, so the volume of the cone is V = 1/3 A * h = 1/3 * 20 m^2 * 9 m = 60 m^3.
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RESPONSE --> With this problem you have to take 1/3 and divide it by the number that you get when you multiply the base and heighth together. In this instance you would have 20 times 9 which equals 180. Then you would take 180 and divide it by 1/3 to get an answer of 60m^3. I added a bunch of extra things in there when I shouldn't have. When I went back and reworked the problem I realized what I did and understand the right way to do it. self critique assessment: 3
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16:36:24 `q008. What is a volume of a sphere whose radius is 4 meters?
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RESPONSE --> The volume is 5.3m^3. confidence assessment: 2
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16:40:07 The volume of a sphere is V = 4/3 pi r^3, where r is the radius of the sphere. In this case r = 4 m so V = 4/3 pi * (4 m)^3 = 4/3 pi * 4^3 m^3 = 256/3 pi m^3.
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RESPONSE --> I understand now that you have to take the 4 and place it over 3 to get 4/3 and then you have to cube 4 to get 64. Then you multiply 4/3 and 64 together to get 85.3 and then turn the decimal into a fraction to get 256/3 pi m^3. self critique assessment: 3
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16:43:44 `q009. What is the volume of a planet whose diameter is 14,000 km?
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RESPONSE --> The volume is 5600/3pi km^3. confidence assessment: 2
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16:50:05 The planet is presumably a sphere, so to the extent that this is so the volume of this planet is V = 4/3 pi r^3, where r is the radius of the planet. The diameter of the planet is 14,000 km so the radius is half this, or 7,000 km. It follows that the volume of the planet is V = 4/3 pi r^3 = 4/3 pi * (7,000 km)^3 = 4/3 pi * 343,000,000,000 km^3 = 1,372,000,000,000 / 3 * pi km^3. This result can be approximated to an appropriate number of significant figures.
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RESPONSE --> I undserstand that the planet is like a sphere. I needed to half the diameter of 14,000 to get a radius of 7,000 so that it could be cubed. I then took that nuber and multiplied it to 4/3. It then gives me numbers in scienctific notation. self critique assessment: 2
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16:52:10 `q010. Summary Question 1: What basic principle do we apply to find the volume of a uniform cylinder of known dimensions?
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RESPONSE --> You multiply the base times the height. The cm or m is cubed because you usually have a squared number already there. You just need to multiply the numbers to find the answer. confidence assessment: 2
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16:52:58 The principle is that when the cross-section of an object is constant, its volume is V = A * h, where A is the cross-sectional area and h the altitude. Altitude is measure perpendicular to the cross-section.
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RESPONSE --> To me the altitude is like the base and then you have the height. They are multilpied together to give you the answer that you are looking for. self critique assessment: 2
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16:54:18 `q011. Summary Question 2: What basic principle do we apply to find the volume of a pyramid or a cone?
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RESPONSE --> You take 1/3 of the base and height. You multiply the base times the height. What number you get there you then multiply it by 1/3 to get you final answer. This answe is usually in cm or m cubed. confidence assessment: 2
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16:55:13 The volumes of these solids are each 1/3 the volume of the enclosing figure. Each volume can be expressed as V = 1/3 A * h, where A is the area of the base and h the altitude as measured perpendicular to the base.
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RESPONSE --> Since you are using 1/3 you just multiply the numbers that you have together and then multiply them by 1/3 to get the final answer. self critique assessment: 2
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16:55:57 `q012. Summary Question 3: What is the formula for the volume of a sphere?
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RESPONSE --> The formula is V=4/3 pi r^3. confidence assessment: 2
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16:56:51 The volume of a sphere is V = 4/3 pi r^3, where r is the radius of the sphere.
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RESPONSE --> You are given the radius and you have to cube the radius and then multiply that number by 4/3 to get the final answer in cm or m cubed. self critique assessment: 2
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16:57:56 `q013. Explain how you have organized your knowledge of the principles illustrated by the exercises in this assignment.
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RESPONSE --> As I went through the problems I wrote them down drew a little picture to help me see it better. I then worked it out step by step to make sure that I was on the right track. I then would see if the answer made since to me. I made notes along the way to help me with future references. confidence assessment: 3
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