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.
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.
Daniel is planning to buy his first house. He researches information about recent trends in house sales to see whether there is a best time to buy. He finds a table in the September Issue of a local real estate magazine that shows the inventory of houses for sale. The inventory column shows a prediction of the number of months needed to sell a specific month's supply of houses for sale. The table also shows the median sales price for houses each month.
Daniel wonders whether housing prices are more likely to increase or decrease in any special month. If he randomly selects a month other than January from the table, what is the price as a fraction, that the median sales price in that month was an increase over the previous month?
Correct Answer & Rationale
Correct Answer: 4\7
To determine the fraction of months where the median sales price increased over the previous month, one must analyze the data presented in the table. The correct answer, 4/7, indicates that out of the seven months considered (excluding January), there were four months where prices rose compared to the month prior. Other options, such as 3/7 or 5/7, misrepresent the data by either underestimating or overestimating the actual increases. A fraction of 3/7 would imply that only three months saw an increase, which contradicts the evidence. Similarly, 5/7 would suggest an unrealistic majority of months experienced price hikes, not aligning with the data. Thus, 4/7 accurately reflects the observed trends in the provided data.
To determine the fraction of months where the median sales price increased over the previous month, one must analyze the data presented in the table. The correct answer, 4/7, indicates that out of the seven months considered (excluding January), there were four months where prices rose compared to the month prior. Other options, such as 3/7 or 5/7, misrepresent the data by either underestimating or overestimating the actual increases. A fraction of 3/7 would imply that only three months saw an increase, which contradicts the evidence. Similarly, 5/7 would suggest an unrealistic majority of months experienced price hikes, not aligning with the data. Thus, 4/7 accurately reflects the observed trends in the provided data.
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.
Read the phrase below.
the quotient of three less than a number and six more than four times a number
Which expression is equivalent to this phrase?
- A. (3-x)/(4x + 6)
- B. (x - 3)(4x + 6)
- C. (x-3)/(4x + 6)
- D. 4x - 3 + 6
Correct Answer & Rationale
Correct Answer: C
The phrase describes a mathematical expression involving a number, denoted as \( x \). "Three less than a number" translates to \( x - 3 \), while "six more than four times a number" translates to \( 4x + 6 \). Therefore, the entire expression is the quotient of these two parts, resulting in \( \frac{x - 3}{4x + 6} \), which matches option C. Option A incorrectly suggests a subtraction in the numerator, altering the intended expression. Option B implies multiplication instead of division, misrepresenting the relationship. Option D presents a simplified expression rather than a quotient, which does not align with the original phrase.
The phrase describes a mathematical expression involving a number, denoted as \( x \). "Three less than a number" translates to \( x - 3 \), while "six more than four times a number" translates to \( 4x + 6 \). Therefore, the entire expression is the quotient of these two parts, resulting in \( \frac{x - 3}{4x + 6} \), which matches option C. Option A incorrectly suggests a subtraction in the numerator, altering the intended expression. Option B implies multiplication instead of division, misrepresenting the relationship. Option D presents a simplified expression rather than a quotient, which does not align with the original phrase.