course MTH 174 1:18 p.m. 6/3/10 If your solution to stated problem does not match the given solution, you should self-critique per instructions at
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Given Solution: `aIf 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. Self-critique (if necessary): OK Self-critique rating #$&* ********************************************* Question: `q006. If a rectangular solid made of a uniform, homogeneous material has dimensions 4 cm by 7 cm by 12 cm and if its mass is 700 grams then what is its density in grams per cubic cm? Your solution: V= 4 cm * 7 cm* 12 cm= 336 cm^3 then d= m/V= 700 g/ 336 cm^3= 2.08 g/cm^3 confidence rating #$&* 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Given Solution: `aThe 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 (for example 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). NOTE TO STUDENT: (in this note the instructor attempts to clarify the idea of 'demonstrating what you do and do not understand about the statement of the problem' and 'giving a phrase-by-phrase analysis of the given solution') You did not respond to the question and did not self-critique. You would be expected to address the question, stating what you do and do not understand. • For example you should understand what a rectangular solid with dimensions 4 cm by 7 cm by 12 cm is, and how to find its volume and surface area. You might not know what to do with this information (for example you might well not understand that it's the volume and not the surface area that's related to density), but from previous work you should understand this much, and should at least mention something along the lines of 'well, I do know that I can find the volume and/or surface area of that solid' in a partial solution. • The word 'density' is clearly very important. Even if you don't know what density is, you could note from the statement of the problem that its units here are said to be 'grams per cubic centimeter'. Having noted these things, you will be much better prepared to understand the information in the given solution. Then you need to address the information in the given solution. A 'phrase-by-phrase' analysis is generally very beneficial: • I expect you understand the first statement from previous knowledge (you should have this understanding from prerequisite courses, and if not you encountered it in the preceding 'volumes' exercise): 'The volume of this solid is 4 cm * 7 cm * 12 cm = 336 cm^3.' It would of course be appropriate to ask a question here if necessary. • It is likely that, as is the case with many students, the concept of density is not that familiar to you. However if this wasn't addressed specifically in prerequisite courses, those courses would be expected to prepare you to understand this concept. The statement 'Its density in grams per cm^3 is the number of grams in each cm^3.' serves as a definition of density. In your self-critique you should have addressed what what this phrase means to you, and what you do or do not understand about it • The next phrase is 'We find this quantity by dividing the number of grams by the number of cm^3.' You would be expected to understand that this phrase is related to the preceding, and as best you can to address the connection. At this point many students would need to ask a question, and it would be perfectly appropriate to do so (or to have done so regarding previous statements). • The subsequent phrase 'density = 700 grams / (336 cm^3) = 2.06 grams / cm^3' is an illustration of the ideas and definitions in the preceding statements. A reasonable self-critique would demonstrate your attempt to understand this statement and its connection to the preceding. Once again questions would also be appropriate and welcome. • The above addresses sufficient information to solve the problem. If you get to this point, you're probably doing OK and you wouldn't necessarily be expected to address the rest of the given solution, which expands on the finer details of the problem and provides additional information. The basic prerequisite courses should have prepared you to understand the information, but students entering Liberal Arts Mathematics, College Algebra and even Precalculus or Applied Calculus (or Physics 121-122) courses probably don't need to address anything beyond the basic solution at this point. Though Precalculus and Applied Calculus students could benefit from doing so, and if time permits would certainly be encouraged to do so, time is also a factor and it would be understandable if these students chose to move on. • Students entering the Mth 173-4 sequence or the Phy 201-202 or 231-232 sequence would be expected to either completely understand all the details of the given solution, or address them in your self-critique. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating #$&* 3 ********************************************* Question: `q007. What is the mass of a sphere of radius 4 meters if its average density is 3,000 kg/cubic meter? Your solution: Here d= 3,000 kg/m^3 and V= 4/3*π*r^2= 4/3*π*4^3 m= 256/3 π m^3. Rearranging d= m/V to m= V*d and substituting gives m= 3,000 kg/m^3 * 256/3 π m^3 = 256000π kg= 2.56x10^5 π kg ≈ 8.04x10^5 kg Confidence Assessment: 3 Given Solution: `aA average density of 3000 kg / cubic meter implies that, at least on the average, every cubic meter has a mass of 3000 kg. So to find the mass of the sphere we multiply the number of cubic meters by 3000 kg. The volume of a sphere of radius 4 meters is 4/3 pi r^3 = 4/3 * pi (4m)^3 = 256/3 * pi m^3. So the mass of this sphere is mass = density * volume = 256 / 3 * pi m^3 * 3000 kg / m^3 = 256,000 * pi kg. This result can be approximated to an appropriate number of significant figures. Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating #$&* 3 ********************************************* Question: `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? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: d1= 4 g/cm^3, V1= 6 cm^3, so m1= 24 g. d2= 2 g/cm^3, V2= 10 cm^3 so m2= 20 g. Average density= total mass/total volume= (24 g + 20 g)/(10 cm^3 + 6 cm^3) = 2.75 g/cm^3 Confidence Assessment: 3
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Given Solution: `aThe 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. Self-critique (if necessary): OK Self-critique rating #$&* 3 ********************************************* Question: `q009. In a large box of dimension 2 meters by 3 meters by 5 meters we place 27 cubic meters of sand whose density is 2100 kg/cubic meter, surrounding a total of three cubic meters of cannon balls whose density is 8,000 kg per cubic meter. What is the average density of the material in the box? Your solution: V= 30 m^3, m1= 2100 kg/m^3 * 27 m^3= 56700 kg. m2= 8000 kg/m^3 * 3 m^3= 2400 kg. Average density= (m1+m2)/total V= 80700 kg/ 30 m^3= 2690 kg/m^3 Confidence Assessment: 3 Given Solution: `aWe find the average density from the total mass and the total volume. The mass of the sand is 27 m^3 * 2100 kg / m^3 = 56,700 kg. The mass of the cannonballs is 3 m^3 * 8,000 kg / m^3 = 24,000 kg. The average density is therefore average density = total mass / total volume = (56,700 kg + 24,000 kg) / (27 m^3 + 3 m^3) = 80,700 kg / (30 m^3) = 2,700 kg / m^3, approx.. Self-critique (if necessary): OK Self-critique rating #$&* 3 ********************************************* Question: `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? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: This one is timely! V= 1,700,000 m^2 * 0.015 m= 25500 m^3 and m= d*V= 860 kg/m^3 * 25500 m^3= 21,930,000 kg = 2.2x10^7 kg. Confidence Assessment: 3 Given Solution: `aThe 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 = 21 930 000 kg. This result should be rounded according to the number of significant figures in the given information. STUDENT QUESTION I didn’t round to the most significant figure. ???? How important is this? INSTRUCTOR RESPONSE It will be important. This document is preliminary; the issue of significant figures will be addressed more specifically as we move into the course. Right now I just want you to be aware of the general idea. Self-critique (if necessary): I rounded. Self-critique rating #$&* 3 ********************************************* Question: `q011. Part 1 Summary Question 1: How do we find the surface area of a cylinder? Your solution: Add the area of the rectangle 'wrapped' around the cylinder, A=b*h= 2πr*h, to the area of the top and bottom circles, 2(π*r62), for the total surface area. confidence rating #$&* 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aThe 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. Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating #$&* 3 ********************************************* Question: `q012. Part 1 Summary Question 2: What is the formula for the surface area of a sphere? Your solution: Spherical surface area= 4*π*r^2. Confidence Assessment: 3 Given Solution: `aThe surface area of a sphere is A = 4 pi r^2, where r is the radius of the sphere. Self-critique (if necessary): OK Self-critique rating #$&* 3 ********************************************* Question: `q013. Part 1 Summary Question 3: What is the meaning of the term 'density'. Your solution: Density is mass per unit volume or how much mass is in any give volume of the object. This can be exact or average. Confidence Assessment:3 Given Solution: `aThe 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' Self-critique (if necessary): Ok Self-critique rating #$&* 3 ********************************************* Question: `q014. Part 1 Summary Question 4: If we know average density and mass, how can we find volume? Your solution: Volume can be found by rearranging d=m/V to V= m/d or divided the mass by the density. confidence rating #$&* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Given Solution: `aSince mass = ave density * volume, it follows by simple algebra that volume = mass / ave density. Self-critique (if necessary): I didn't say average. Self-critique rating #$&*3 ********************************************* Question: `q015. Part 1 Explain how you have organized your knowledge of the principles illustrated by the exercises in this assignment. This is also basic geometry ideas necessary for all math classes. "