Factor the expression completely: 45bcx - 10ax
- A. 5x(9bc - 2a)
- B. 5(9bc - 2a)
- C. x(45bc - 10a)
- D. 5x(9bc + 2a)
Correct Answer & Rationale
Correct Answer: A
To factor the expression 45bcx - 10ax completely, we start by identifying the greatest common factor (GCF). The GCF of the coefficients 45 and 10 is 5, and both terms contain the variable x. Thus, we can factor out 5x, resulting in 5x(9bc - 2a). Option A accurately reflects this factorization. Option B lacks the variable x, which is essential in the original expression. Option C incorrectly factors out only x, missing the GCF of 5. Option D alters the sign of the second term, which does not represent the original expression correctly.
To factor the expression 45bcx - 10ax completely, we start by identifying the greatest common factor (GCF). The GCF of the coefficients 45 and 10 is 5, and both terms contain the variable x. Thus, we can factor out 5x, resulting in 5x(9bc - 2a). Option A accurately reflects this factorization. Option B lacks the variable x, which is essential in the original expression. Option C incorrectly factors out only x, missing the GCF of 5. Option D alters the sign of the second term, which does not represent the original expression correctly.
Other Related Questions
Type your answer in the box. You may use numbers, a decimal point (-), and/or a negative sign (-) in your answer.
A truck driver sees a road sign warning of an 8% road incline. To the nearest tenth of a foot, what will be the change in the truck's vertical position, in feet, during the time it takes the truck's horizontal position to change by 1 mile? (1 mile = 5,280 ft)
Correct Answer & Rationale
Correct Answer: 422.4
To determine the vertical change during a 1-mile horizontal distance on an 8% incline, we calculate the vertical rise using the formula: vertical rise = incline percentage × horizontal distance. Here, 8% as a decimal is 0.08, and the horizontal distance is 5,280 feet. Therefore, the vertical change is 0.08 × 5,280 = 422.4 feet. Other options are incorrect as they either miscalculate the incline percentage or the conversion of miles to feet. For instance, values significantly lower than 422.4 feet suggest a misunderstanding of the incline's impact, while options above this value imply an overestimation of the incline's effect on vertical change.
To determine the vertical change during a 1-mile horizontal distance on an 8% incline, we calculate the vertical rise using the formula: vertical rise = incline percentage × horizontal distance. Here, 8% as a decimal is 0.08, and the horizontal distance is 5,280 feet. Therefore, the vertical change is 0.08 × 5,280 = 422.4 feet. Other options are incorrect as they either miscalculate the incline percentage or the conversion of miles to feet. For instance, values significantly lower than 422.4 feet suggest a misunderstanding of the incline's impact, while options above this value imply an overestimation of the incline's effect on vertical change.
Which graph represents a function?
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A.
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B.
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C.
Correct Answer & Rationale
Correct Answer: B
To determine which graph represents a function, we apply the Vertical Line Test. This test states that if a vertical line intersects the graph at more than one point, the graph does not represent a function. Option A fails this test, as a vertical line can intersect the graph at multiple points, indicating it is not a function. Option C also does not satisfy the criteria, showing multiple intersections with vertical lines. In contrast, Option B passes the Vertical Line Test, as any vertical line drawn will intersect the graph at only one point, confirming it represents a function.
To determine which graph represents a function, we apply the Vertical Line Test. This test states that if a vertical line intersects the graph at more than one point, the graph does not represent a function. Option A fails this test, as a vertical line can intersect the graph at multiple points, indicating it is not a function. Option C also does not satisfy the criteria, showing multiple intersections with vertical lines. In contrast, Option B passes the Vertical Line Test, as any vertical line drawn will intersect the graph at only one point, confirming it represents a function.
For an emergency service call, a plumbing company charges a flat fee of $60 plus $40 an hour. A customer remembers paying at least $200 for an emergency service. Which phrase describes the number of hours the plumbing company was at the service call?
- A. at most 2 hours
- B. at most 3.5 hours
- C. at least 3.5 hours
- D. at least 2 hours
Correct Answer & Rationale
Correct Answer: C
To determine the number of hours the plumbing company was on the service call, we start with the total charge of at least $200. The charge consists of a flat fee of $60 plus $40 per hour. First, subtract the flat fee from the total: $200 - $60 = $140. Next, divide this by the hourly rate: $140 ÷ $40 = 3.5 hours. This indicates that the service lasted at least 3.5 hours. Option A (at most 2 hours) is incorrect, as 2 hours would only cost $140. Option B (at most 3.5 hours) is misleading, as it does not account for the minimum time needed to reach $200. Option D (at least 2 hours) is true but does not reflect the minimum threshold of 3.5 hours. Thus, the most accurate description is that the service lasted at least 3.5 hours.
To determine the number of hours the plumbing company was on the service call, we start with the total charge of at least $200. The charge consists of a flat fee of $60 plus $40 per hour. First, subtract the flat fee from the total: $200 - $60 = $140. Next, divide this by the hourly rate: $140 ÷ $40 = 3.5 hours. This indicates that the service lasted at least 3.5 hours. Option A (at most 2 hours) is incorrect, as 2 hours would only cost $140. Option B (at most 3.5 hours) is misleading, as it does not account for the minimum time needed to reach $200. Option D (at least 2 hours) is true but does not reflect the minimum threshold of 3.5 hours. Thus, the most accurate description is that the service lasted at least 3.5 hours.
((5^3 * 2^4)^2)(5^(-2) * 2^5)
- A. 5^3 * 2^11
- B. 5^(-12) * 2^40
- C. 5^4 * 2^13
- D. (-5)^8 * 2^13
Correct Answer & Rationale
Correct Answer: C
To simplify the expression \(((5^3 * 2^4)^2)(5^{-2} * 2^5)\), first apply the power of a product rule. This gives \(5^{6} * 2^{8}\) from the first part. Next, combine this with the second part, \(5^{-2} * 2^{5}\). Adding the exponents for the base 5: \(6 + (-2) = 4\). For base 2: \(8 + 5 = 13\). Thus, the final expression simplifies to \(5^4 * 2^{13}\). Option A is incorrect as it miscalculates the exponents. Option B has incorrect exponents and signs. Option D introduces an unnecessary negative sign and does not match the simplified expression.
To simplify the expression \(((5^3 * 2^4)^2)(5^{-2} * 2^5)\), first apply the power of a product rule. This gives \(5^{6} * 2^{8}\) from the first part. Next, combine this with the second part, \(5^{-2} * 2^{5}\). Adding the exponents for the base 5: \(6 + (-2) = 4\). For base 2: \(8 + 5 = 13\). Thus, the final expression simplifies to \(5^4 * 2^{13}\). Option A is incorrect as it miscalculates the exponents. Option B has incorrect exponents and signs. Option D introduces an unnecessary negative sign and does not match the simplified expression.