course Mth 174 This ends the third assignment. ϐђ{Dݫ
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15:47:14 `q001. There are 10 questions and 5 summary questions in this assignment. What is surface area of a rectangular solid whose dimensions are 3 meters by 4 meters by 6 meters?
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RESPONSE --> The surface area of a rectangular solid is calculated with SA= 2(lh) + 2(lw) + 2(wh) SA= 2 (3 * 4) + 2 (3 * 6) + 2(6 * 4) SA=24 + 36 + 48 SA= 108m confidence assessment: 2
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15:48:30 A rectangular solid has six faces (top, bottom, front, back, left side, right side if you're facing it). The pairs top and bottom, right and left sides, and front-back have identical areas. This solid therefore has two faces with each of the following dimensions: 3 m by 4 m, 3 m by 6 m and 4 m by 6 m, areas 12 m^2, 18 m^2 and 24 m^2. Total area is 2 * 12 m^2 + 2 * 18 m^2 + 2 * 24 m^2 = 108 m^2.
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RESPONSE --> self critique assessment: 3
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15:55:38 `q002. What is the surface area of the curved sides of a cylinder whose radius is five meters and whose altitude is 12 meters? If the cylinder is closed what is its total surface area?
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RESPONSE --> To find the surface area of an entire cylinder you use SA=2pi r^2 + 2pi rh The area of just the curved sides would be SA=2pi rh SA=2pi (5)(12) SA=2pi 60 SA=120pi The total area would be SA= 2pi 5^2 + 2pi (5)(12) SA=2pi 25 + 2pi 60 SA= 50pi + 120pi SA= 170pi m confidence assessment: 1
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15:58:17 The circumference of this cylinder is 2 pi r = 2 pi * 5 m = 10 pi m. If the cylinder was cut by a straight line running up its curved face then unrolled it would form a rectangle whose length and width would be the altitude and the circumference. The area of the curved side is therefore A = circumference * altitude = 10 pi m * 12 m = 120 pi m^2. If the cylinder is closed then it has a top and a bottom, each a circle of radius 5 m with resulting area A = pi r^2 = pi * (5 m)^2 = 25 pi m^2. The total area would then be total area = area of sides + 2 * area of base = 120 pi m^2 + 2 * 25 pi m^2 = 170 pi m^2.
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RESPONSE --> I should think of the curved side as a rolled up rectangle and solve the surface area by multiplying the circumference by the height of the cylinder. confidence assessment: 2
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16:01:32 `q003. What is surface area of a sphere of diameter three cm?
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RESPONSE --> To find surface area of a sphere you use SA= 4pi r^2. The radius is 1.5 (half the diameter) so: SA=4 pi 1.5^2 SA=4pi 2.25 SA=9pi cm^2 confidence assessment: 2
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16:01:42 The surface area of a sphere of radius r is A = 4 pi r^2. This sphere has radius 3 cm / 2, and therefore has surface area A = 4 pi r^2 = 4 pi * (3/2 cm)^2 = 9 pi cm^2.
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RESPONSE --> self critique assessment: 3
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16:04:31 `q004. What is hypotenuse of a right triangle whose legs are 5 meters and 9 meters?
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RESPONSE --> To find the hypotenuse of a right triangle, use a^2 + b^2 = c^2, where a and b are the legs and c is the hypotenuse so: 9^2 + 5^2= c^2 81 + 25 = c^2 106=c^2 the square root of 106 = 10.30 so c=10.30 confidence assessment: 2
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16:05:02 The Pythagorean Theorem says that the hypotenuse c of a right triangle with legs a and b satisfies the equation c^2 = a^2 + b^2. So, since all lengths are positive, we know that c = sqrt(a^2 + b^2) = sqrt( (5 m)^2 + (9 m)^2 ) = sqrt( 25 m^2 + 81 m^2) = sqrt( 106 m^2 ) = 10.3 m, approx.. Note that this is not what we would get if we made the common error of assuming that sqrt(a^2 + b^2) = a + b; this would tell us that the hypotenuse is 14 m, which is emphatically not so. There is no justification whatsoever for applying a distributive law (like x * ( y + z) = x * y + x * z ) to the square root operator.
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RESPONSE --> self critique assessment: 3
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16:07:46 `q005. If the hypotenuse of a right triangle has length 6 meters and one of its legs has length 4 meters what is the length of the other leg?
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RESPONSE --> a^2 + b^2= c^2 a^2 + 4^2=6^2 a^2 +16 = 36 a^2= 20 the square root of 20 is 4.5 so a=4.5m confidence assessment: 3
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16:07:54 If c is the hypotenuse and a and b the legs, we know by the Pythagorean Theorem that c^2 = a^2 + b^2, so that a^2 = c^2 - b^2. Knowing the hypotenuse c = 6 m and the side b = 4 m we therefore find the unknown leg: a = sqrt( c^2 - b^2) = sqrt( (6 m)^2 - (4 m)^2 ) = sqrt(36 m^2 - 16 m^2) = sqrt(20 m^2) = sqrt(20) * sqrt(m^2) = 2 sqrt(5) m, or approximately 4.4 m.
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RESPONSE --> self critique assessment: 3
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16:13:02 `q006. If a rectangular solid made of a uniform, homogeneous material has dimensions 4 cm by 7 cm by 12 cm and if its mass is 700 grams then what is its density in grams per cubic cm?
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RESPONSE --> Density is calculated by mass / volume so first you have to find the volume. V= l*w*h V=4*7*12 V=336cm^3 D= 700 / 336 D= 2.08 g/cm^3 confidence assessment: 2
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16:15:31 The volume of this solid is 4 cm * 7 cm * 12 cm = 336 cm^3. Its density in grams per cm^3 is the number of grams in each cm^3. We find this quantity by dividing the number of grams by the number of cm^3. We find that density = 700 grams / (336 cm^3) = 2.06 grams / cm^3. Note that the solid was said to be uniform and homogeneous, meaning that it's all made of the same material, which is uniformly distributed. So each cm^3 does indeed have a mass of 2.06 grams. Had we not known that the material was uniform and homogeneous we could have said that the average density is 2.06 grams / cm^3, but not that the density is 2.06 grams / cm^3 (the object could be made of two separate substances, one with density less than 2.06 grams / cm^3 and the other with density greater than 2.06 g / cm^3, in appropriate proportions; neither substance would have density 2.06 g / cm^3, but the average density could be 2.06 g / cm^3).
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RESPONSE --> You would say the average density when working with a figure that is not made out of the same material because different substances may have higher or lower densities. self critique assessment: 2
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16:26:38 `q007. What is the mass of a sphere of radius 4 meters if its average density is 3,000 kg/cubic meter?
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RESPONSE --> You can use the density equation of a sphere to find mass also. First you have to convert kg to g and 3000kg = 9,000,000g AD= mass / 4/3 pi r^3 9,000,000 = Mass / 4/3 pi 4^3 9,000,000 = Mass/ 4/3 pi 64 9,000,000 = Mass/ 48pi 9,000,000 * 48pi = Mass Mass = 432,000,000pi g/m^3 confidence assessment: 1
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16:29:01 A average density of 3000 kg / cubic meter implies that, at least on the average, every cubic meter has a mass of 3000 kg. So to find the mass of the sphere we multiply the number of cubic meters by 3000 kg. The volume of a sphere of radius 4 meters is 4/3 pi r^3 = 4/3 * pi (4m)^3 = 256/3 * pi m^3. So the mass of this sphere is mass = density * volume = 256 / 3 * pi m^3 * 3000 kg / m^3 = 256,000 * pi kg. This result can be approximated to an appropriate number of significant figures.
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RESPONSE --> I thought when working with density and mass you had to use grams. Also I multiplied by 3/4 instead of 4/3. self critique assessment: 2
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16:31:37 `q008. If we build a an object out of two pieces of material, one having a volume of 6 cm^3 at a density of 4 grams per cm^3 and another with a volume of 10 cm^3 at a density of 2 grams per cm^3 then what is the average density of this object?
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RESPONSE --> The average density is 3g/cm^3 because it is the average of both materials. confidence assessment: 1
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16:33:18 The first piece has a mass of 4 grams / cm^3 * 6 cm^3 = 24 grams. The second has a mass of 2 grams / cm^3 * 10 cm^3 = 20 grams. So the total mass is 24 grams + 20 grams = 44 grams. The average density of this object is average density = total mass / total volume = (24 grams + 20 grams) / (6 cm^3 + 10 cm^3) = 44 grams / (16 cm^3) = 2.75 grams / cm^3.
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RESPONSE --> Now I know to use total mass / total volume to find average density. self critique assessment: 2
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16:41:14 `q009. In a large box of dimension 2 meters by 3 meters by 5 meters we place 27 cubic meters of sand whose density is 2100 kg/cubic meter, surrounding a total of three cubic meters of cannon balls whose density is 8,000 kg per cubic meter. What is the average density of the material in the box?
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RESPONSE --> Average Density = total mass / total volume To find the mass of the sand and balls you have to use D= m/v for each and then add the masses together. Sand= 2100 = m/27 m= 56700 Balls= 8000= m/3 m=24000 total mass = 80700kg total volume=30m^3 AD= 80700 / 30 AD=2690kg/cm^3 confidence assessment: 2
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16:41:38 We find the average density from the total mass and the total volume. The mass of the sand is 27 m^3 * 2100 kg / m^3 = 56,700 kg. The mass of the cannonballs is 3 m^3 * 8,000 kg / m^3 = 24,000 kg. The average density is therefore average density = total mass / total volume = (56,700 kg + 24,000 kg) / (27 m^3 + 3 m^3) = 80,700 kg / (30 m^3) = 2,700 kg / m^3, approx..
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RESPONSE --> self critique assessment: 3
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16:45:48 `q010. How many cubic meters of oil are there in an oil slick which covers 1,700,000 square meters (between 1/2 and 1 square mile) to an average depth of .015 meters? If the density of the oil is 860 kg/cubic meter the what is the mass of the oil slick?
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RESPONSE --> 1,700,000 * .015 = 25000 m^3 of oil D= m / v 860 = m / 25000 860m = 25000 m= 29.65kg confidence assessment: 2
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16:48:09 The volume of the slick is V = A * h, where A is the area of the slick and h the thickness. This is the same principle used to find the volume of a cylinder or a rectangular solid. We see that the volume is V = A * h = 1,700,000 m^2 * .015 m = 25,500 m^3. The mass of the slick is therefore mass = density * volume = 860 kg / m^3 * 24,400 m^3 = 2,193,000 kg. This result should be rounded according to the number of significant figures in the given information.
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RESPONSE --> I put the mass in the volumes place and therefore divided when I should have multiplied. self critique assessment: 2
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16:48:43 `q011. Summary Question 1: How do we find the surface area of a cylinder?
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RESPONSE --> SA= 2pi r^2 + 2pi rh confidence assessment: 3
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16:49:02 The curved surface of the cylinder can be 'unrolled' to form a rectangle whose dimensions are equal to the circumference and the altitude of the cylinder, so the curved surface has volume Acurved = circumference * altitude = 2 pi r * h, where r is the radius and h the altitude. The top and bottom of the cylinder are both circles of radius r, each with resulting area pi r^2. {]The total surface area is therefore Acylinder = 2 pi r h + 2 pi r^2.
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RESPONSE --> self critique assessment: 3
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16:49:19 `q012. Summary Question 2: What is the formula for the surface area of a sphere?
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RESPONSE --> SA= 4pi r^2 confidence assessment: 3
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16:49:28 The surface area of a sphere is A = 4 pi r^2, where r is the radius of the sphere.
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RESPONSE --> self critique assessment: 3
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16:50:36 `q013. Summary Question 3: What is the meaning of the term 'density'.
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RESPONSE --> Density is how much mass is contained in a given unit volume confidence assessment:
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16:50:47 The average density of an object is its mass per unit of volume, calculated by dividing its total mass by its total volume. If the object is uniform and homogeneous then its density is constant and we can speak of its 'density' as opposed to its 'average density'
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RESPONSE --> self critique assessment:
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16:51:36 `q014. Summary Question 4: If we know average density and mass, how can we find volume?
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RESPONSE --> If you know the average density and mass you can find volume by multiplying the density and mass. confidence assessment: 3
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16:52:09 Since mass = ave density * volume, it follows by simple algebra that volume = mass / ave density.
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RESPONSE --> self critique assessment: 3
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16:52:36 `q015. Explain how you have organized your knowledge of the principles illustrated by the exercises in this assignment.
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RESPONSE --> I have organized my knowledge of the principles illustrated here by writing down all the problems, sloving them and correcting answers. confidence assessment: 3
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