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.
Other Related Questions
One online movie-streaming service costs $8 per month and charges $1.50 per movie. A second service costs $2 per month and charges $2 per movie. For what number of movies per month is the monthly cost of both services the same?
- A. 3
- B. 6
- C. 5
- D. 12
- E. 20
Correct Answer & Rationale
Correct Answer: D
To determine when the costs of both services are equal, we can set up equations based on the monthly fees and per-movie charges. For the first service: Cost = $8 + $1.50 * number of movies (m) Cost = $8 + 1.5m For the second service: Cost = $2 + $2 * number of movies (m) Cost = $2 + 2m Setting the two equations equal gives us: $8 + 1.5m = $2 + 2m Rearranging leads to: $6 = 0.5m m = 12 Thus, when 12 movies are rented, the costs are equal. Options A (3), B (6), and C (5) yield different costs, as they do not satisfy the equation. Option E (20) results in a higher cost for the second service, confirming that 12 is the only solution where both services cost the same.
To determine when the costs of both services are equal, we can set up equations based on the monthly fees and per-movie charges. For the first service: Cost = $8 + $1.50 * number of movies (m) Cost = $8 + 1.5m For the second service: Cost = $2 + $2 * number of movies (m) Cost = $2 + 2m Setting the two equations equal gives us: $8 + 1.5m = $2 + 2m Rearranging leads to: $6 = 0.5m m = 12 Thus, when 12 movies are rented, the costs are equal. Options A (3), B (6), and C (5) yield different costs, as they do not satisfy the equation. Option E (20) results in a higher cost for the second service, confirming that 12 is the only solution where both services cost the same.
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.
Through which pair of points could a line of best fit be drawn for the data on the scatterplot?
- A. (0, 36) and (11, 74)
- B. (1, 39) and (6, 60)
- C. (5, 50) and (6, 60)
- D. (6, 60) and (8, 60)
- E. (8, 60) and (11, 74)
Correct Answer & Rationale
Correct Answer: A
Option A, with points (0, 36) and (11, 74), shows a significant range in both x and y values, indicating a strong upward trend that aligns well with the overall direction of the data. Option B, while showing an upward trend, has a narrower range and may not represent the overall data as effectively. Option C features two points that are too close together, limiting their ability to define a clear line of best fit. Option D includes points with the same y-value, suggesting a horizontal line that does not capture the data's trend. Option E, like A, has a valid upward trend but does not span the data range as effectively as A.
Option A, with points (0, 36) and (11, 74), shows a significant range in both x and y values, indicating a strong upward trend that aligns well with the overall direction of the data. Option B, while showing an upward trend, has a narrower range and may not represent the overall data as effectively. Option C features two points that are too close together, limiting their ability to define a clear line of best fit. Option D includes points with the same y-value, suggesting a horizontal line that does not capture the data's trend. Option E, like A, has a valid upward trend but does not span the data range as effectively as A.
The volume of 1 cup of water is 14.4 cubic inches. The diameter of an empty cylindrical can is 3.0 inches. The can holds 2.0 cups of water. What is the height of the can, to the nearest 0.1 inch?
- A. 1
- B. 2
- C. 3.1
- D. 4.1
- E. 6.2
Correct Answer & Rationale
Correct Answer: D
To find the height of the can, first determine the total volume of water it holds. Since 1 cup is 14.4 cubic inches, 2 cups equal 28.8 cubic inches (2 x 14.4). The formula for the volume of a cylinder is V = πr²h. The radius (r) of the can is half the diameter: 1.5 inches. Plugging in the values: 28.8 = π(1.5)²h. Calculating the area of the base gives approximately 7.07. Rearranging the equation for height (h) results in h ≈ 4.1 inches. Options A (1), B (2), C (3.1), and E (6.2) do not satisfy the volume calculation, as they yield heights inconsistent with the required volume based on the diameter provided.
To find the height of the can, first determine the total volume of water it holds. Since 1 cup is 14.4 cubic inches, 2 cups equal 28.8 cubic inches (2 x 14.4). The formula for the volume of a cylinder is V = πr²h. The radius (r) of the can is half the diameter: 1.5 inches. Plugging in the values: 28.8 = π(1.5)²h. Calculating the area of the base gives approximately 7.07. Rearranging the equation for height (h) results in h ≈ 4.1 inches. Options A (1), B (2), C (3.1), and E (6.2) do not satisfy the volume calculation, as they yield heights inconsistent with the required volume based on the diameter provided.