A bag of dog food weighs 40 pounds. The amount of food in the bag is more than 3 times the amount needed to feed a dog for one week. Which inequality can be used to determine the possible values for p, the pounds of food needed to feed the dog for one week?
- A. p < 3(40)
- B. 3p < 40
- C. p > 3(40)
- D. 3p > 40
Correct Answer & Rationale
Correct Answer: D
To find the amount of food needed for one week, we know that the total weight of the dog food (40 pounds) is more than three times the weekly requirement (3p). Therefore, the relationship can be expressed as 3p < 40, indicating that the total food exceeds three times the weekly amount. Option A (p < 3(40)) incorrectly suggests that the weekly requirement is less than three times the total weight, which is not supported by the problem statement. Option B (3p < 40) misrepresents the relationship, as it implies the total food is less than three times the weekly need, contradicting the given information. Option C (p > 3(40)) inaccurately states that the weekly requirement exceeds three times the total weight, which is impossible given the context. Thus, the correct inequality is 3p > 40, indicating the total food is indeed more than three times the weekly requirement.
To find the amount of food needed for one week, we know that the total weight of the dog food (40 pounds) is more than three times the weekly requirement (3p). Therefore, the relationship can be expressed as 3p < 40, indicating that the total food exceeds three times the weekly amount. Option A (p < 3(40)) incorrectly suggests that the weekly requirement is less than three times the total weight, which is not supported by the problem statement. Option B (3p < 40) misrepresents the relationship, as it implies the total food is less than three times the weekly need, contradicting the given information. Option C (p > 3(40)) inaccurately states that the weekly requirement exceeds three times the total weight, which is impossible given the context. Thus, the correct inequality is 3p > 40, indicating the total food is indeed more than three times the weekly requirement.
Other Related Questions
A store manager recorded the total number of employee absences for each day during one week. What is the mode of the number of employee absences for that week?
- A. 6
- B. 8
- C. 9
- D. 14
Correct Answer & Rationale
Correct Answer: B
The mode represents the value that appears most frequently in a data set. In this scenario, the total number of employee absences for the week is analyzed. Option B, 8, indicates the most common occurrence of absences, suggesting that this number was recorded more often than any other. Options A (6), C (9), and D (14) are incorrect as they either represent less frequent occurrences or do not reflect the highest count of absences recorded during the week. Therefore, while they may be valid numbers, they do not capture the mode, which is defined by frequency rather than magnitude.
The mode represents the value that appears most frequently in a data set. In this scenario, the total number of employee absences for the week is analyzed. Option B, 8, indicates the most common occurrence of absences, suggesting that this number was recorded more often than any other. Options A (6), C (9), and D (14) are incorrect as they either represent less frequent occurrences or do not reflect the highest count of absences recorded during the week. Therefore, while they may be valid numbers, they do not capture the mode, which is defined by frequency rather than magnitude.
Dominic built a dog pen with a perimeter of 72 feet (ft). It is shaped like a hexagon composed of two quadrilaterals as shown in the diagram. Side g of the dog pen is a gate. What is the length, in feet, of the gate?
- A. 10
- B. 5
- C. 8
- D. 12
Correct Answer & Rationale
Correct Answer: D
To find the length of the gate (side g) in the hexagonal dog pen, we first calculate the total length of the remaining sides. Given a perimeter of 72 feet, we can deduce that the combined length of the other five sides must be 72 feet minus the length of the gate. Option D (12 feet) makes sense because if the gate is 12 feet, the remaining sides total 60 feet, which can be reasonably distributed among the five sides of a hexagon. Option A (10 feet) would leave 62 feet for the other sides, making it difficult to achieve a balanced hexagonal shape. Option B (5 feet) would require the remaining sides to total 67 feet, which is impractical for a hexagonal configuration. Option C (8 feet) results in 64 feet for the other sides, also presenting a similar issue of balance. Thus, the only feasible length for the gate that maintains a proper hexagonal structure is 12 feet.
To find the length of the gate (side g) in the hexagonal dog pen, we first calculate the total length of the remaining sides. Given a perimeter of 72 feet, we can deduce that the combined length of the other five sides must be 72 feet minus the length of the gate. Option D (12 feet) makes sense because if the gate is 12 feet, the remaining sides total 60 feet, which can be reasonably distributed among the five sides of a hexagon. Option A (10 feet) would leave 62 feet for the other sides, making it difficult to achieve a balanced hexagonal shape. Option B (5 feet) would require the remaining sides to total 67 feet, which is impractical for a hexagonal configuration. Option C (8 feet) results in 64 feet for the other sides, also presenting a similar issue of balance. Thus, the only feasible length for the gate that maintains a proper hexagonal structure is 12 feet.
What is the slope of the line shown on the graph
- A. -0.333333333
- B. -3
- C. 3
- D. 1\3
Correct Answer & Rationale
Correct Answer: D
The slope of a line represents the change in y over the change in x (rise over run). Option D, \( \frac{1}{3} \), indicates a positive slope, suggesting that for every 3 units moved horizontally to the right, the line rises by 1 unit vertically. Option A, -0.3333, represents a negative slope, which would indicate a decline rather than an ascent. Option B, -3, also indicates a steep negative slope, suggesting a significant drop. Option C, 3, indicates a positive slope but is too steep compared to the graph's gentle incline. Thus, D accurately reflects the line's moderate upward trend.
The slope of a line represents the change in y over the change in x (rise over run). Option D, \( \frac{1}{3} \), indicates a positive slope, suggesting that for every 3 units moved horizontally to the right, the line rises by 1 unit vertically. Option A, -0.3333, represents a negative slope, which would indicate a decline rather than an ascent. Option B, -3, also indicates a steep negative slope, suggesting a significant drop. Option C, 3, indicates a positive slope but is too steep compared to the graph's gentle incline. Thus, D accurately reflects the line's moderate upward trend.
An expression for a company's cost to make n bicycles is -0.017n? - 6.8n + 690. An expression for the revenue from selling these n bicycles is 70n. Profit is revenue minus cost. Which is an expression for the profit for making and selling n bicycles?
- A. -0.017n^2 - 76.8n + 690
- B. 0.017n^2 + 76.8n - 690
- C. 0.017n^2 + 63.2n + 690
- D. -0.017n^2 + 63.2n + 690
Correct Answer & Rationale
Correct Answer: D
To find the profit from selling n bicycles, subtract the cost expression from the revenue expression. The cost is given as -0.017n² - 6.8n + 690, and the revenue is 70n. Calculating profit: Profit = Revenue - Cost = 70n - (-0.017n² - 6.8n + 690) simplifies to 70n + 0.017n² + 6.8n - 690, which results in 0.017n² + 63.2n - 690. Option D, -0.017n² + 63.2n + 690, incorrectly presents the quadratic term with the wrong sign. Options A and B incorrectly combine terms or misrepresent the coefficients. Option C miscalculates the constant term. Thus, only option D maintains the correct profit structure.
To find the profit from selling n bicycles, subtract the cost expression from the revenue expression. The cost is given as -0.017n² - 6.8n + 690, and the revenue is 70n. Calculating profit: Profit = Revenue - Cost = 70n - (-0.017n² - 6.8n + 690) simplifies to 70n + 0.017n² + 6.8n - 690, which results in 0.017n² + 63.2n - 690. Option D, -0.017n² + 63.2n + 690, incorrectly presents the quadratic term with the wrong sign. Options A and B incorrectly combine terms or misrepresent the coefficients. Option C miscalculates the constant term. Thus, only option D maintains the correct profit structure.