Which of the following expressions is equivalent to: 6x³ + 7x² + 1/x?
- A. 63 + 72 + 1/x
- B. 63 + 72 + 1
- C. 6x² + 7x + 1/x
- D. 6x² + 7x + 1
- E. 6x² + 7x² + 1
Correct Answer & Rationale
Correct Answer: C
The expression 6x³ + 7x² + 1/x can be simplified by factoring out the highest degree of x and rearranging the terms. Option C, 6x² + 7x + 1/x, contains the correct coefficients for the x terms, but with the degrees adjusted appropriately. Option A incorrectly suggests a constant sum of 63 and 72, which does not relate to the original expression. Option B also misrepresents the original expression by omitting the variable terms entirely. Option D fails to maintain the degree of x in the cubic term, while option E mistakenly combines the x² terms incorrectly, resulting in an inaccurate expression.
The expression 6x³ + 7x² + 1/x can be simplified by factoring out the highest degree of x and rearranging the terms. Option C, 6x² + 7x + 1/x, contains the correct coefficients for the x terms, but with the degrees adjusted appropriately. Option A incorrectly suggests a constant sum of 63 and 72, which does not relate to the original expression. Option B also misrepresents the original expression by omitting the variable terms entirely. Option D fails to maintain the degree of x in the cubic term, while option E mistakenly combines the x² terms incorrectly, resulting in an inaccurate expression.
Other Related Questions
Let g(x) = x². What is the average rate of change of the function from x = 4 to x = 8?
- A. 1/12
- B. $2
- C. $4
- D. $12
- E. $48
Correct Answer & Rationale
Correct Answer: C
To determine the average rate of change of the function g(x) = x² from x = 4 to x = 8, we use the formula: (g(b) - g(a)) / (b - a), where a = 4 and b = 8. Calculating g(4) = 4² = 16 and g(8) = 8² = 64. Thus, the average rate of change is (64 - 16) / (8 - 4) = 48 / 4 = 12. Option A (1/12) is incorrect as it underestimates the change. Option B ($2) and Option D ($12) miscalculate the average rate. Option E ($48) represents the total change but does not account for the interval length. The correct average rate of change is $12, reflecting the consistent increase of the function over the specified interval.
To determine the average rate of change of the function g(x) = x² from x = 4 to x = 8, we use the formula: (g(b) - g(a)) / (b - a), where a = 4 and b = 8. Calculating g(4) = 4² = 16 and g(8) = 8² = 64. Thus, the average rate of change is (64 - 16) / (8 - 4) = 48 / 4 = 12. Option A (1/12) is incorrect as it underestimates the change. Option B ($2) and Option D ($12) miscalculate the average rate. Option E ($48) represents the total change but does not account for the interval length. The correct average rate of change is $12, reflecting the consistent increase of the function over the specified interval.
An irrigation pivot makes a circle with a radius of about 400 meters. Which of the following values is closest to the area, in square meters, of the circle?
- A. 1300
- B. 2500
- C. 160000
- D. 502700
- E. 1579100
Correct Answer & Rationale
Correct Answer: D
To find the area of a circle, the formula \( A = \pi r^2 \) is used, where \( r \) is the radius. With a radius of 400 meters, the area calculates to approximately \( A = \pi \times (400)^2 \approx 502700 \) square meters, making option D the closest value. Option A (1300) is far too low, indicating a misunderstanding of the formula. Option B (2500) is also significantly underestimated for such a large radius. Option C (160000) is closer but still incorrect, as it neglects the multiplication by \( \pi \). Option E (1579100) overestimates the area, suggesting a miscalculation of the radius or the area formula.
To find the area of a circle, the formula \( A = \pi r^2 \) is used, where \( r \) is the radius. With a radius of 400 meters, the area calculates to approximately \( A = \pi \times (400)^2 \approx 502700 \) square meters, making option D the closest value. Option A (1300) is far too low, indicating a misunderstanding of the formula. Option B (2500) is also significantly underestimated for such a large radius. Option C (160000) is closer but still incorrect, as it neglects the multiplication by \( \pi \). Option E (1579100) overestimates the area, suggesting a miscalculation of the radius or the area formula.
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.
Which of the following statements is true about the graphs of f(x) = x and g(x) = 3x in the standard (x, y) coordinate plane?
- A. The graphs will not intersect.
- B. The graphs will intersect only at the point (0,0).
- C. The graphs will intersect only at the point (0,1).
- D. The graphs will intersect only at the point (1,1).
- E. The graphs will intersect only at the point (3,3).
Correct Answer & Rationale
Correct Answer: D
The graphs of f(x) = x and g(x) = 3x represent two linear functions with different slopes. The first function has a slope of 1, while the second has a slope of 3. They will intersect where their outputs are equal, which occurs when x = 1, resulting in the point (1,1). Option A is incorrect as the lines, being linear, will intersect at some point. Option B is misleading; they intersect at (0,0) but also at (1,1). Option C is false because g(1) = 3, not 1. Option E is incorrect since g(3) = 9, not 3. Thus, the only valid intersection point is (1,1).
The graphs of f(x) = x and g(x) = 3x represent two linear functions with different slopes. The first function has a slope of 1, while the second has a slope of 3. They will intersect where their outputs are equal, which occurs when x = 1, resulting in the point (1,1). Option A is incorrect as the lines, being linear, will intersect at some point. Option B is misleading; they intersect at (0,0) but also at (1,1). Option C is false because g(1) = 3, not 1. Option E is incorrect since g(3) = 9, not 3. Thus, the only valid intersection point is (1,1).