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Surface Areas |
reviews the meaning, reasoning and calculation of surface areas of some common geometric figures | the meaning of surface area, reasoning about surface area, some important formulas for surface areas |
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If your solution to stated problem does not match the given solution, you should self-critique per instructions at
http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm
.
Your solution, attempt at solution. If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.
003. Misc: Surface Area, Pythagorean Theorem, Density
Question: `q001. There are 17 questions in this document.
What is surface area of a rectangular solid whose dimensions
are 3 meters by 4 meters by 6 meters?
Your solution:
Confidence Assessment:
Given Solution:
`aA 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.
Self-critique (if necessary):
Self-critique Rating:
Question:
`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?
Your solution:
Confidence Assessment:
Given Solution:
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.
Self-critique (if necessary):
Self-critique Rating:
Question:
`q003. What is surface
area of a sphere of diameter three cm?
Your solution:
Confidence Assessment:
Given Solution:
`aThe 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.
NOTE TO STUDENT:
While your work on most problems has been good, you left this problem blank and didn't self-critique.
You should self-critique here.
For example you should acknowledge having made note of the formula for the surface area of the sphere, which I expect you didn't know before.
I expect from your previous answers that you are very capable of applying the formula once you have it, and based on this history you probably wouldn't need to self-critique that aspect of the process.
Self-critique (if necessary):
Self-critique Rating:
Question:
`q004. What is
hypotenuse of a right triangle whose legs are 5 meters and 9 meters?
Your solution:
Confidence Assessment:
Given Solution:
`aThe 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.
Self-critique (if necessary):
Self-critique Rating:
Question:
`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?
Your solution:
Confidence Assessment:
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):
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:
Confidence Assessment:
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):
Self-critique Rating:
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:
Confidence Assessment:
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):
Self-critique Rating:
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?
Your solution:
Confidence Assessment:
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):
Self-critique Rating:
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:
Confidence Assessment:
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):
Self-critique Rating:
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?
Your solution:
Confidence Assessment:
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 * 25,500 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):
Self-critique Rating:
Question:
`q011.
Part 1 Summary Question
1: How do we find the surface area of a
cylinder?
Your solution:
Confidence Assessment:
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 area
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):
Self-critique Rating:
Question:
`q012.
Part 1 Summary Question
2: What is the formula for the surface
area of a sphere?
Your solution:
Confidence Assessment:
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):
Self-critique Rating:
Question:
`q013.
Part 1 Summary Question
3: What is the meaning of the term 'density'.
Your solution:
Confidence Assessment:
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):
Self-critique Rating:
Question:
`q014.
Part 1 Summary Question
4: If we know average density and mass,
how can we find volume?
Your solution:
Confidence Assessment:
Given Solution:
Since mass = ave density * volume, it follows by simple
algebra that volume = mass / ave density.
Self-critique (if necessary):
Self-critique Rating:
Question:
`q015.
Part 1 Explain how you
have organized your knowledge of the principles illustrated by the exercises in
this assignment.
Your solution:
Confidence Assessment:
Question: `q016. The hypotenuse of a right triangle is also the diameter of a certain circle. If the legs of the triangle are 4 feet and 9 feet, what is the area of the circle?
Optional question (somewhat challenging):
The hypotenuse of a right triangle is also the diameter of a certain circle.
The legs of the triangle are also
Your solution:
Confidence Assessment:
Self-critique Rating:
Question: `q017. A hemisphere of radius 10 cm is filled with a water whose density is 1 gram per cm^3. A cone with base radius 10 cm and altitude 20 cm is filled with oil whose density is 0.8 grams / cm^3. The contents of the two are poured into a container and shaken in an attempt to mix them. What is the average density of the mix?
Your solution:
Confidence Assessment:
Self-critique Rating: