Questions and Answers
What is the definition of a cylinder?
Write an equation for a line with slope 2 passing through the point (−3, 7).
y = 2x - 1
Rewrite LOG2(1/2) = -1 as an exponential equation.
2^{-1} = 1/2
Calculate the distance between the points (2,−2) and (7,−6): d = √((x2-x1)^2 + (y2-y1)^2). The answer is ____.
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What type of line will be used to graph the inequality |v+6| > 2?
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What is the length of a training track formed by a rectangle and two semicircles with rectangle dimensions 82m long and 57m wide?
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What are the solutions to the equation |2u + 10| = 4?
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Write the equations for the downloads of standard and high-quality versions of a song given total downloads and total size.
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How many liters of natural oil are needed to make 864 liters of Petrolyn oil with a ratio of 5 liters of natural oil for every 7 liters of synthetic oil?
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It is NOT a function if x is repeated more than ____. NOT a function if it fails the _____ test.
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What is the conversion factor to convert 90° to radians?
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What is the formula for the area of a circle?
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Find the remaining credit on a phone card after multiple calls given the remaining credit after certain minutes.
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What is the perimeter formula for a rectangle?
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Find the slope of a line if given two points.
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Study Notes
Geometry and Algebra Concepts
- A cylinder is a three-dimensional shape with two parallel circular bases connected by a curved surface.
- The line equation can be written as (y = mx + b) where (m) is the slope. For a line passing through (-3, 7) with a slope of 2, the equation is (y = 2x - 1).
- Rewrite logarithmic equations as exponential equations; for example, (\log_2(1/2) = -1) means (2^{-1} = \frac{1}{2}).
- To calculate the distance between points (2, -2) and (7, -6), use the distance formula (d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}), yielding an exact distance of (5).
- Graph inequalities on a number line, using a dotted line for strict inequalities (e.g., (|v + 6| > 2)) and a solid line for inclusive inequalities.
Mathematical Applications
- A training field consists of a rectangle and two semicircles, with a total perimeter calculation that combines all sides and curves.
- Determine the number of downloads of a song by setting up a system of equations based on total downloads and file sizes.
- For Petrolyn motor oil, with a ratio of natural to synthetic oil, calculate required amounts for a total volume.
- A function fails if any (x) value is repeated. The vertical line test can visually confirm this.
Measurement Conversions
- To convert degrees to radians, multiply by (\frac{\pi}{180}). For example, (90^\circ) becomes (\frac{\pi}{2}) radians.
- For converting (9\pi/4) to degrees, multiply by (\frac{180}{\pi}).
Financial Mathematics
- Determine regular prices based on sale percentages. A table sold for $284 is 71% of its regular price.
- If the sofa sale price is $211.20 with a discount of 67%, find the original price using (OP = \frac{211.20}{0.33}).
Factorization and Algebraic Manipulation
- For expressions like (17v^4 + 27v^3 / 3v^3), simplify to (17v + 9) through division.
- To factor quadratic expressions such as (16x^2 - 8x - 3), identify pairs of numbers multiplying to the product while summing to a middle coefficient.
- Utilize (p^{-3} / p^{-2} = \frac{1}{p}) for exponent rules.
Circle, Area, and Volume Calculations
- Area of a circle: (A = \pi r^2). The circumference can be calculated as (C = \pi d).
- The volume of a cylinder follows the formula: (Volume = \pi r^2 h).
- The area of a triangle is calculated using (A = \frac{1}{2}bh) and for a parallelogram (A = bh).
Inequalities and Functions
- Graph inequalities such as (y > -\frac{4}{3}x - 4) by rearranging to slope-intercept form and shading the appropriate region.
- Identify domain restrictions by factoring expressions in the denominator and determining the values that make it zero.
Vertex Form and Graphing
- Complete the square to convert to vertex form (y = a(x-h)^2 + k) for parabolic expressions.
- Understand that a parabola opens upward or downward depending on the sign of the leading coefficient.
Final Notes
- Make use of inverse sine and cosine functions to find angle measurements when necessary.
- The proper method for calculating areas with mixed geometric shapes involves combining standard area formulas with adjustments for semicircles.
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Description
Test your knowledge of key geometry and algebra concepts, including geometric shapes, line equations, logarithmic transformations, and distance calculations. This quiz also covers inequalities and their graphical representations. Dive into various mathematical applications related to these topics!
