Solved by verified expert:10 questions…The other three documents are just examples and methods my professor want us to follow to follow.
math_012_quiz_1_spring_2018__1_.docx
graphing_grid.docx
direct_and_inverse_variation_applications.pdf
how_to_create_a_number_line_solution_set_from_your_keyboard.pdf
steps_for_solving_a_linear_equation___math012.pdf
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Math 012 Quiz 1
Spring 2018
Professor: Dr. Kate Bauer
Name________________________________
Instructions:
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The quiz is worth 50 points. There are 10 problems, each worth 5 points. Your score on
the quiz will be converted to a percentage and posted in your assignment folder with
comments.
This quiz is open book and open notes, and you may take as long as you like on it
provided that you submit the quiz no later than the due date posted in our course schedule
of the syllabus. You may refer to your textbook, notes, and online classroom materials,
but you may not consult anyone.
You must show all of your work to receive full credit. If a problem does not seem to
require work, write a sentence or two to justify your answer.
Please type your work in your copy of the quiz, or if you prefer, create a document
containing your work. Scanned work is also acceptable. Be sure to include your name in
the document. Review instructions for submitting your quiz in the Quizzes Module.
If you have any questions, please contact me by e-mail (kate.bauer@umuc.edu).
At the end of your quiz you must include the following dated statement with your name typed in lieu of
a signature. Without this signed statement you will receive a zero.
I have completed this quiz myself, working independently and not consulting anyone except the instructor. I
have neither given nor received help on this quiz.
Name:
Date:
Please remember to show ALL of your work on every problem. Read the basic
rules for showing work below BEFORE you start working on the quiz.
a) Each step should show the complete expression or equation rather than a piece of it.
b) Each new step should follow logically from the previous step, following rules of algebra.
c) Each new step should be beneath the previous step.
d) The equal sign, =, should only connect equal numbers or expressions.
If you have questions about showing work, please ask.
Math 012
Quiz 1
Page 2
Did you read the rules for showing work on page 1 of the quiz? If not, please go back to page
1 now and read them. If you do not show work correctly, you will not earn full credit.
1) Solve the equation below. Show all work following the methods discussed in class; if an
equation includes fractions, clear the fractions in the first step. If the equation has a unique
solution, please show the complete check of your answer.
7 x 3 24 8 x
2) Solve the equation below. Show all work following the methods discussed in class; if an
equation includes fractions, clear the fractions in the first step. If the equation has a unique
solution, please show the complete check of your answer.
15 3 y 33 y 1 4 3 y 6
Math 012
Quiz 1
Page 3
3) Solve the equation below. Show all work following the methods discussed in class; if an
equation includes fractions, clear the fractions in the first step. If the equation has a unique
solution, please show the complete check of your answer.
3 8 2 x 8 x 4 x 2
4) Solve the equation below. Show all work following the methods discussed in class; if an
equation includes fractions, clear the fractions in the first step. If the equation has a unique
solution, please show the complete check of your answer.
7 y 5 y9
5
4
2
Math 012
Quiz 1
Page 4
5) Solve the equation below. Show all work following the methods discussed in class; if an
equation includes fractions, clear the fractions in the first step. If the equation has a unique
solution, please show the complete check of your answer.
1 y 23 5 y 1
2 12 3
Math 012
Quiz 1
Page 5
6) The area of an oil spill on land varies directly as the volume of oil spilled. If an 80-gallon oil
spill covers an area of 180 square feet, how many square feet would a 200-gallon oil spill cover?
Please include the following in your work:
a) A statement defining any unknown quantities in the problem in terms of variables.
b) An equation involving the variable(s) corresponding to the information given in the
problem.
c) A complete solution to the equation, showing all work.
d) A complete answer to the question asked, including appropriate units.
7) In kick boxing, the energy needed to break a board of standard width and thickness varies
inversely with the length of the board. If 3.2 Joules of energy are required to break an 27-inch
board, how many Joules of energy are required to break a 24-inch board?
Please include the following in your work:
a) A statement defining any unknown quantities in the problem in terms of variables.
b) An equation involving the variable(s) corresponding to the information given in the
problem.
c) A complete solution to the equation, showing all work.
d) A complete answer to the question asked, including appropriate units.
Math 012
Quiz 1
Page 6
8) A plumber cut a 40-foot piece of pipe into two pieces. The length of the longer piece was
eight feet more than three times the length of the shorter piece. What was the length of each
piece of pipe?
Please include the following in your work:
a) A statement defining any unknown quantities in the problem in terms of variables.
b) An equation involving the variable(s) corresponding to the information given in the
problem.
c) A complete solution to the equation, showing all work.
d) A complete answer to the question asked, including appropriate units.
9) Bob left Andrew’s house and drove toward the ocean. One hour later, Andrew left, driving
12 mph faster than Bob in an effort to catch up with him. After three hours, Andrew caught up
with Bob. What was each man’s average rate?
Please include the following in your work:
a) A statement defining any unknown quantities in the problem in terms of variables.
b) An equation involving the variable(s) corresponding to the information given in the
problem.
c) A complete solution to the equation, showing all work.
d) A complete answer to the question asked, including appropriate units.
Math 012
Quiz 1
Page 7
10) Becky would like to have at least $240,000 saved for her daughter’s college education. If
she invests $95,000 in an education account paying 6.15% interest compounded quarterly, will
she reach her goal in 18 years? Show all work to justify your answer and include appropriate
units. The final amount in the account should be rounded to the nearest cent.
Please include the following in your work:
a) A statement defining any unknown quantities in the problem in terms of variables.
b) An equation involving the variable(s) corresponding to the information given in the
problem.
c) A complete solution to the equation, showing all work.
d) A complete answer to the question asked, including appropriate units.
End of quiz: please remember to sign and date the honor statement in the box on the first page
of the quiz.
y
x
Direct and Inverse Variation Applications
We see examples of these mathematical concepts every day in our lives. “The higher I set the
thermostat, the higher my gas bill will be,” is an example of direct variation. In direct variation
two quantities are directly related, in this case the marking on the thermostat and the resulting
gas bill: as one quantity goes up, the other quantity goes up.
An example of inverse variation from the real world is “the bigger the car, the lower the gas
mileage.” In inverse variation, two quantities are related in this way: as one quantity goes up, the
other quantity goes down.
In both kinds of variation problems, the two quantities have a constant of variation relating them.
Mathematicians commonly use the letter “k” to stand for that constant.
If y varies directly as x, we write:
y
y
kx
=
which can also be written
k
x
If y varies inversely as x, we write:
y
k
=
which can also be written yx k
x
Direct Variation Example: the dosage of a certain baby medication varies directly as the weight
of the baby. If a baby who weighs 12 pounds should be given 80 mg of the medication, how
many milligrams should a baby who weighs 15 pounds be given?
Step 1: write the general equation of variation. We will let D = dosage and W = weight.
D = kW
Step 2: use the information given to find the value of the constant of variation, k.
80= k ⋅12
80
=k
12
20
k=
3
Step 3: write the particular equation of variation with the value of k in place.
D=
20
W
3
Step 4: use the particular equation to answer the question asked.
20
W
3
20
D = (15 )
3
= 100
D=
According to this model, a 15-pound baby should be given 100 mg of the medication.
Inverse Variation Example: the weight of an object varies inversely as the square of the
object’s distance from the center of the Earth. If a person weighs 180 pounds on the Earth’s
surface, how much will he weigh 10 miles above the surface of the Earth? (Note: the radius of
the Earth is about 4000 miles).
Step 1: Write the general variation equation, using k as the unknown constant of
variation. Let’s let W = weight and M = number of miles from the center of the Earth.
Here is our general equation:
W=
k
M2
Notice the M 2 in the denominator. We square the M in this problem because the weight
varies inversely as the SQUARE of the distance from Earth’s center.
Step 2: Use the information given in the problem to find the constant of variation.
k
M2
k
180 =
(The weight is 180 on the Earth’s surface.)
40002
k
180 =
16000000
180*16, 000, 000 = k
W=
=
k 2,880, 000, 000 or 2.88 ×109
Note: the second version of the value of k is written in scientific notation, which we will
discuss during Week 4. For a preview, see Section 5.3 of our text.
Step 3: Write the particular equation for this problem by putting the constant of variation
in place.
W=
2.88 ×109
M2
Step 4: Answer the question asked. In this problem we were asked what the weight of
this person would be 10 miles above the Earth’s surface. A distance of 10 miles above
the Earth’s surface is 4010 miles from the Earth’s center.
2.88 ×109
40102
2.88 ×109
=
16, 080,100
W=
2.88 ×109
=
1.60801×107
= 179.103
At 10 miles above the Earth’s surface, our 180-pound friend only weighs 179.103
pounds!
For more examples of direct and inverse variation, please see Section 1.7 of our text.
How to create a number line solution set from your keyboard
We often graph the solution set to an inequality on a number line. You may be
wondering how we can create these kinds of graphs from the keyboard. Let’s look
together at some examples.
Example 1:
2x −1 < 5
2x −1 +1 < 5 +1
2x < 6
2x 6
<
2 2
x<3
In the example above, the solution set, x < 3, means that x can be any real number less
than 3. The inequality sign is strict, which means the 3 itself is not included in the
solution set.
In some textbooks, the number line solution set would look like this:
The open circle at 3 indicates that 3 is not included in the solution set. The blue arrow
pointing left from 3 indicates that x can be any real number less than 3.
The standard way to show the same solution set on a number line created from the
keyboard is below:
x<3
< =====================)--------------- >
3
Notice that we use ordinary keys to create this graph: the < and > keys for the arrows at
each end of the number line, the repeated equal sign, =====, for the part of the number
line that IS the solution set, the repeated hyphen signs, ——, for the part of the number
line that is NOT the solution set, and the parenthesis mark in place of the open circle.
Number lines
2
If the solution set were x < 3 instead of x < 3, we would see a filled-in circle on the
number line in many textbooks. When we type the number line, we use a bracket to
indicate an “or-equal-to” inequality:
x<3
< ====================]--------------- >
3
Example 2:
4 − 3 x < 16
4 − 4 − 3 x < 16 − 4
−3 x < 12
−3 x 12
>
−3 −3
x > −4
Note: in this example, we have to apply Part 2
of the Multiplication Property of Inequality
because we are dividing both sides of the
inequality by a negative number.
The number line graph of this strict inequality is below:
x > −4
< ------------------(========== >
-4
Example 3:
2 ( x + 6)
≤ x −5
3
2 ( x + 6)
3
≤ 3 [ x − 5]
3
2 ( x + 6 ) ≤ 3 ( x − 5)
2 x + 12 ≤ 3 x − 15
2 x − 3 x + 12 ≤ 3 x − 3 x − 15
−1x + 12 ≤ −15
−1x + 12 − 12 ≤ −15 − 12
−1x ≤ −27
−1x −27
≥
−1
−1
x ≥ 27
Note: here is it again! Part 2 of the
Multiplication Property of Inequality
must be applied whenever we multiply
or divide both sides of an inequality by
a negative number.
Number lines
3
The number line graph of this “or-equal-to” inequality is below:
x ≥ 27
< ------------------[============ >
27
Sometimes we work with compound inequalities, for example: −4 < x ≤ 2 . This means
“-4 is less than x and x is less than or equal to 2.” A textbook version of the number line
might look like this:
We can show the same solution set from the computer keyboard in this way:
< --------(=====================]------------- >
-4
2
Notice that we replace the open circle at -4 with a parenthesis and the filled-in circle at 2
with a bracket. The repeated = signs indicate that the solutions lie between -4 and 2. The
bracket tells us that 2 is included in the solution set and the parenthesis tells us that -4 is
not included.
Steps for Solving a Linear Equation
Step 1: If an equation contains fractions or decimals, multiply both sides by the LCD to
clear the equation of fractions or decimals.
Step 2: Use the Distributive Property to remove parentheses if they are present.
Step 3: Simplify each side of the equation by combining like terms.
Step 4: Get all variable terms on one side and all numbers on the other side by using
the addition property of equality.
Step 5: Get the variable alone by using the multiplication property of equality.
Step 6: Check the solution by substituting it into the original equation.
Special Linear Equations:
There are two special kinds of equations that do not lead to unique solutions. In both
cases, when we follow the steps listed above, we find that the variables drop out of the
equation. In one case, we are left with an identity, a statement like 5 = 5, which is
always true. In that case, the solution set for the equation is the set of all real numbers.
In the other case, we are left with a contradiction, a statement like 4 = 7, which is never
true. In that case we state the conclusion that “there is no solution to the equation.”
For an example of each of these special cases, see Section 1.3 of our text.
Note about checking answers:
Checking an answer to an equation is always a good idea, and will generally be expected
in this course. When we check an answer, we substitute it into the original equation,
and then work on each side of that equation, simplifying as much as possible, until the
two sides are identical. When we reach an identity, we conclude that our answer is
correct. If we don’t reach an identity, we know that there is an error in the work of
finding the answer, in the check, or in both, and we try every step again.
When checking, we work in the original equation only. There are two reasons for this.
First, by checking in the original equation, we avoid repeating any errors we might have
made in solving the equation. Second, when we check an answer, we are testing for
equality, and therefore we cannot apply any properties of equality while we check. We
cannot, for example, add something to both sides of the equation or multiply both sides
by the LCD when we perform the check. We work ONLY with the original equation when
we perform the check.
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