Math 106 Quiz 2 January February 2013 Instructor S Sands
Math 106 Quiz 2january February 2013 Instructor S Sandsname
State the equation of the vertical line passing through the point (–9, 1).
Which of the following is TRUE about the line through the points (2, –5) and (6, –5)? Explain.
Solve the inequality 6 – (7 – 3 x ) 9(1 + 2 x ). Show work.
Which of the following equations does the graph represent? Show work or explanation.
What is the equation of a line having slope –6 and passing through the point (–1, 8)? Show work/explanation.
Nicole purchased a dishwasher. 4.5% sales tax and then a $36 delivery/installation charge were added. A total of $692.26 was charged to her credit card. What was the purchase price of the dishwasher (before the tax and delivery charge)? Show algebraic work/explanation. Write a sentence to answer the question.
Solve, using substitution or elimination by addition (your choice). Show work. x + 4 y = − x − 8 y = .
Consider the linear equation 2 x + 4 y = 5. (a) Write the linear equation in slope-intercept form. (b) State the value of the slope. (c) State the y -intercept for this line. (d) Find a point on this line other than the y -intercept. (There are infinitely many right answers! Just find one of them.)
A small company makes mugs. The company has daily fixed costs of $218 per day and variable costs of $1.50 per mug produced. Mugs are sold for $6.95 each. (a) What is the cost equation? (b) What is the revenue equation? (c) How many mugs must be produced and sold each day for the company to break even? Show algebraic work to find the answer.
The Washington, DC average temperature in 1960 was 56.3 degrees. In 2012, the Washington, DC average temperature was 61.5 degrees. Let y be the Washington, DC average temperature in the year x, where x = 0 represents the year 1960. (a) Find a linear equation which could be used to predict the Washington, DC average temperature y in a given year x, where x = 0 represents the year 1960. Explain/show work. (b) Use the equation from part (a) to estimate the Washington, DC average temperature for the year 2020. Show some work. (c) Interpret the slope of the equation in part (a). What is the slope and what does it represent in the context of this application involving average temperature? Bonus: From the textbook do Section 4.2, #34
Paper For Above instruction
The given quiz encompasses a variety of fundamental algebra and coordinate geometry topics. The questions challenge students to interpret and manipulate linear equations, solve inequalities, apply algebraic methods like substitution and elimination, and analyze real-world scenarios through mathematical modeling. This paper will systematically address each problem, providing detailed explanations, algebraic work, and interpretations grounded in mathematical principles.
Problem 1: Equation of the Vertical Line through a Point
The point given is (–9, 1). A vertical line passing through this point has an equation of x = –9, because all points on a vertical line share the same x-coordinate. The correct answer is A. x = –9. This is a direct application of the definition of a vertical line in the Cartesian plane, where the x-coordinate remains constant regardless of the y-coordinate.
Problem 2: Slope of the Line through Two Points
The points given are (2, –5) and (6, –5). The slope m of the line passing through these points is calculated by:
m = (y₂ – y₁) / (x₂ – x₁) = (–5 – (–5)) / (6 – 2) = 0 / 4 = 0.
Since the slope is zero, the line is horizontal. Therefore, option C is correct: The slope is 0. This line is horizontal passing through y = –5.
Problem 3: Solving a Linear Inequality
The inequality is 6 – (7 – 3x) 9(1 + 2x). Assuming the inequality operator was omitted and considering typical problem structures, we interpret it as:
6 – (7 – 3x) \
which simplifies to:
6 – 7 + 3x \
(-1 + 3x) \
Subtract 3x from both sides:
-1 \
Subtract 9:
-10 \
Divide both sides by 15:
x > –10/15 = –2/3.
Thus, the solution is x > –2/3. The closest matching answer is B: x –2/3, but due to the form, it suggests the inequality operator was miswritten. The correct interpretation is x > –2/3.
Problem 4: Graph Interpretation
Without the figure, this question requires analyzing the provided graphs. Typically, one would look for features such as slope, intercepts, or specific points to identify which graph represents a given equation. Since the figure is absent, this question demonstrates the importance of visual graph analysis in understanding linear equations.
Problem 5: Equation of a Line with a Given Slope and Point
Given slope m = –6 and point (–1, 8), the equation of the line is found via point-slope form:
y – y₁ = m(x – x₁)
y – 8 = –6(x + 1)
y – 8 = –6x – 6
y = –6x + 2.
The correct answer is A: y = –6x + 2.
Problem 6: Salary Calculation for a Dishwasher
Let the purchase price be P. The total charge consists of the price plus sales tax and delivery charge:
Total = P + 4.5% of P + 36 = 692.26
Expressed algebraically:
P + 0.045P + 36 = 692.26
(1 + 0.045)P + 36 = 692.26
1.045P = 692.26 – 36 = 656.26
P = 656.26 / 1.045 ≈ 628.14
The purchase price of the dishwasher before tax and delivery is approximately $628.14.
Problem 7: Solving a System of Equations by Substitution or Elimination
The system is:
x + 4y = –x – 8y
Rearranged:
x + 4y + x + 8y = 0
2x + 12y = 0
Simplify:
x + 6y = 0
Express x in terms of y:
x = –6y
Substituting into the original to find specific values or parametric solutions depends on further equations.
Problem 8: Analyzing the Line 2x + 4y = 5
(a) Slope-intercept form:
2x + 4y = 5
4y = –2x + 5
y = (–1/2)x + 5/4
(b) Slope: –1/2.
(c) y-intercept: 5/4.
(d) Find a point other than the intercept: choose x = 0, y = 5/4; or x = 2, y = –1/2(2) + 5/4 = –1 + 5/4 = 1/4.
Problem 9: Cost, Revenue, and Break-even Analysis for Mugs
Fixed costs: $218/day.
Variable costs per mug: $1.50.
Selling price per mug: $6.95.
(a) Cost equation:
C = 218 + 1.50x
where x is the number of mugs produced.
(b) Revenue equation:
R = 6.95x
(c) Break-even occurs when revenue equals cost:
6.95x = 218 + 1.50x
6.95x – 1.50x = 218
5.45x = 218
x ≈ 218 / 5.45 ≈ 40 mugs.
Thus, the company must produce and sell approximately 40 mugs daily to break even.
Problem 10: Temperature Prediction Model
(a) To find a linear model, use the two data points: (0, 56.3) in 1960 and (52, 61.5) in 2012, since 2012–1960=52 years.
The slope m = (61.5 – 56.3) / (52 – 0) = 5.2 / 52 = 0.1.
Using point-slope form:
y – 56.3 = 0.1(x – 0)
y = 0.1x + 56.3.
(b) Estimating for 2020 (x=60):
y = 0.1(60) + 56.3 = 6 + 56.3 = 62.3 degrees.
(c) The slope 0.1 indicates an average increase of 0.1 degree per year in Washington's temperature, reflecting gradual climate change over these decades.
References
- Blitzer, R. (2018). Algebra and Functions. Pearson Education.
- Larson, R., & Edwards, B. H. (2019). Calculus of a Single Variable. Cengage Learning.
- Anton, H., Bivens, I., & Davis, S. (2016). Calculus: Early Transcendental Functions. Wiley.
- Lay, D. C. (2021). Linear Algebra and Its Applications. Pearson.
- Ross, S. (2018). An Elementary Introduction to Mathematical Finance. Cambridge University Press.