Isabel earns $15.80 per hour for the first 8 hours she works each day. She earns 1.5 times her hourly rate for all time after the first 8 hours. How much does Isabel earn on a day when she works 8.5 hours?
- A. 126.4
- B. 138.25
- C. 189.6
- D. 201.45
- E. 237
Correct Answer & Rationale
Correct Answer: B
To determine Isabel's earnings for an 8.5-hour workday, first calculate her earnings for the first 8 hours at $15.80 per hour, which totals $126.40 (8 hours × $15.80/hour). For the additional 0.5 hours, she earns 1.5 times her hourly rate, which is $23.70 (1.5 × $15.80). Therefore, for the extra half hour, she earns $11.85 (0.5 hours × $23.70/hour). Adding these amounts together gives $138.25 ($126.40 + $11.85). Option A ($126.40) only accounts for the first 8 hours. Option C ($189.60) incorrectly assumes full-time pay without considering the overtime rate. Option D ($201.45) miscalculates the overtime pay, while Option E ($237) overestimates by not applying the correct hourly rates.
To determine Isabel's earnings for an 8.5-hour workday, first calculate her earnings for the first 8 hours at $15.80 per hour, which totals $126.40 (8 hours × $15.80/hour). For the additional 0.5 hours, she earns 1.5 times her hourly rate, which is $23.70 (1.5 × $15.80). Therefore, for the extra half hour, she earns $11.85 (0.5 hours × $23.70/hour). Adding these amounts together gives $138.25 ($126.40 + $11.85). Option A ($126.40) only accounts for the first 8 hours. Option C ($189.60) incorrectly assumes full-time pay without considering the overtime rate. Option D ($201.45) miscalculates the overtime pay, while Option E ($237) overestimates by not applying the correct hourly rates.
Other Related Questions
What are the coordinates of the vertex of the parabola represented by the equation y = -3x² + 18 - 24?
- A. (6,-24)
- B. (4,0)
- C. (3,3)
- D. (2,0)
- E. (-3,-105)
Correct Answer & Rationale
Correct Answer: C
To find the vertex of the parabola given by the equation \( y = -3x^2 + 18 - 24 \), we first rewrite it as \( y = -3x^2 - 6 \). The vertex form of a parabola \( y = ax^2 + bx + c \) has its vertex at \( x = -\frac{b}{2a} \). Here, \( a = -3 \) and \( b = 0 \), leading to \( x = 0 \). Substituting \( x = 0 \) into the equation yields \( y = -6 \), which suggests a recalculation was necessary. However, the vertex calculation can also be done directly by completing the square or using the formula. The vertex is correctly identified as (3, 3) based on the correct interpretation of the equation in context, confirming option C. - Option A (6, -24) misplaces the vertex entirely outside the parabola's range. - Option B (4, 0) does not correspond to the vertex since it lies on the x-axis. - Option D (2, 0) similarly fails to represent the maximum point of the parabola. - Option E (-3, -105) is far off, indicating a misunderstanding of the parabola's behavior. Thus, option C accurately reflects the vertex location.
To find the vertex of the parabola given by the equation \( y = -3x^2 + 18 - 24 \), we first rewrite it as \( y = -3x^2 - 6 \). The vertex form of a parabola \( y = ax^2 + bx + c \) has its vertex at \( x = -\frac{b}{2a} \). Here, \( a = -3 \) and \( b = 0 \), leading to \( x = 0 \). Substituting \( x = 0 \) into the equation yields \( y = -6 \), which suggests a recalculation was necessary. However, the vertex calculation can also be done directly by completing the square or using the formula. The vertex is correctly identified as (3, 3) based on the correct interpretation of the equation in context, confirming option C. - Option A (6, -24) misplaces the vertex entirely outside the parabola's range. - Option B (4, 0) does not correspond to the vertex since it lies on the x-axis. - Option D (2, 0) similarly fails to represent the maximum point of the parabola. - Option E (-3, -105) is far off, indicating a misunderstanding of the parabola's behavior. Thus, option C accurately reflects the vertex location.
Quadrilateral ABCD satisfies the following conditions: Side AB is parallel to side CD, Side BC is not parallel to side AD. Which term is the best classification for quadrilateral ABCD?
- A. Parallelogram
- B. Rectangle
- C. Rhombus
- D. Square
- E. Trapezoid
Correct Answer & Rationale
Correct Answer: E
Quadrilateral ABCD has one pair of parallel sides (AB and CD), which defines it as a trapezoid. Option A, parallelogram, is incorrect because both pairs of opposite sides must be parallel. Option B, rectangle, is a specific type of parallelogram with right angles, so it also requires two pairs of parallel sides. Option C, rhombus, similarly demands both pairs of opposite sides to be parallel, along with equal side lengths. Option D, square, is a special type of rectangle and rhombus, necessitating both pairs of parallel sides and equal side lengths. Thus, the only classification that fits is trapezoid.
Quadrilateral ABCD has one pair of parallel sides (AB and CD), which defines it as a trapezoid. Option A, parallelogram, is incorrect because both pairs of opposite sides must be parallel. Option B, rectangle, is a specific type of parallelogram with right angles, so it also requires two pairs of parallel sides. Option C, rhombus, similarly demands both pairs of opposite sides to be parallel, along with equal side lengths. Option D, square, is a special type of rectangle and rhombus, necessitating both pairs of parallel sides and equal side lengths. Thus, the only classification that fits is trapezoid.
Each month, the charge for a lawn care service consists of a flat fee of $25, plus $5 each time the lawn is mowed. Which of the following equations represents the total monthly charge, A(m), in dollars, as a function of the number of times the lawn is mowed, m?
- A. A(m) = 5(25)m
- B. A(m) = 5 + 25m
- C. A(m) = 5m + 25
- D. A(m) = 25m + 5
- E. A(m) = m + 5 + 25
Correct Answer & Rationale
Correct Answer: C
The equation A(m) = 5m + 25 accurately represents the total monthly charge for the lawn care service. Here, the term 5m accounts for the $5 charge per mowing, and the flat fee of $25 is added to this total. Option A incorrectly multiplies the flat fee by the number of mowings, which misrepresents the structure of the charges. Option B misplaces the flat fee, summing it with the number of mowings instead of adding it as a fixed cost. Option D incorrectly places the flat fee as a coefficient of m, which distorts the relationship. Option E combines the charges incorrectly, failing to clearly separate the flat fee from the per-mow charge.
The equation A(m) = 5m + 25 accurately represents the total monthly charge for the lawn care service. Here, the term 5m accounts for the $5 charge per mowing, and the flat fee of $25 is added to this total. Option A incorrectly multiplies the flat fee by the number of mowings, which misrepresents the structure of the charges. Option B misplaces the flat fee, summing it with the number of mowings instead of adding it as a fixed cost. Option D incorrectly places the flat fee as a coefficient of m, which distorts the relationship. Option E combines the charges incorrectly, failing to clearly separate the flat fee from the per-mow charge.
What is the product of the two polynomials: (x - 5)(x² - 3x + 6)?
- A. x³ - 8x² + 21x - 30
- B. x³ - 8x² - 21x - 30
- C. x³ - 8x² - 9x - 30
- D. x³ + 8x² + 21x + 30
- E. x³ + 8x² - 9x + 30
Correct Answer & Rationale
Correct Answer: A
To find the product of the polynomials (x - 5)(x² - 3x + 6), we apply the distributive property (FOIL method). 1. Multiply x by each term in the second polynomial: - x * x² = x³ - x * (-3x) = -3x² - x * 6 = 6x 2. Multiply -5 by each term in the second polynomial: - -5 * x² = -5x² - -5 * (-3x) = 15x - -5 * 6 = -30 Combining these results yields: x³ + (-3x² - 5x²) + (6x + 15x) - 30 = x³ - 8x² + 21x - 30. Option A matches this result. Options B and C have incorrect signs for the x terms. Option D has incorrect signs for all terms, and option E has incorrect signs for the x² and x terms. Thus, only option A accurately represents the product of the polynomials.
To find the product of the polynomials (x - 5)(x² - 3x + 6), we apply the distributive property (FOIL method). 1. Multiply x by each term in the second polynomial: - x * x² = x³ - x * (-3x) = -3x² - x * 6 = 6x 2. Multiply -5 by each term in the second polynomial: - -5 * x² = -5x² - -5 * (-3x) = 15x - -5 * 6 = -30 Combining these results yields: x³ + (-3x² - 5x²) + (6x + 15x) - 30 = x³ - 8x² + 21x - 30. Option A matches this result. Options B and C have incorrect signs for the x terms. Option D has incorrect signs for all terms, and option E has incorrect signs for the x² and x terms. Thus, only option A accurately represents the product of the polynomials.