A carpenter is installing shelves in 2 offices. Each office will have 4 shelves. The wood the carpenter wants to use comes in 6-foot-long boards. Each shelf is 2 ¼ feet long and is constructed from a single board. How many boards does the carpenter need to buy to make the shelves?
- A. 2
- B. 8
- C. 3
- D. 4
Correct Answer & Rationale
Correct Answer: D
To determine how many boards are needed, first calculate the total length of wood required for the shelves. Each office has 4 shelves, and with 2 offices, that totals 8 shelves. Each shelf is 2 ¼ feet long, which equals 2.25 feet. Therefore, the total length required is 8 shelves x 2.25 feet = 18 feet. Each board is 6 feet long. Dividing the total length (18 feet) by the length of each board (6 feet) gives 3 boards. However, since each board can only be used for one shelf, and we can't cut a board to make multiple shelves, we need to round up to the nearest whole number of boards needed, which is 4. - Option A (2 boards) is insufficient for the total length required. - Option B (8 boards) exceeds the necessary amount. - Option C (3 boards) miscalculates the total need based on the cut requirement. Thus, 4 boards are necessary to accommodate all shelves without waste.
To determine how many boards are needed, first calculate the total length of wood required for the shelves. Each office has 4 shelves, and with 2 offices, that totals 8 shelves. Each shelf is 2 ¼ feet long, which equals 2.25 feet. Therefore, the total length required is 8 shelves x 2.25 feet = 18 feet. Each board is 6 feet long. Dividing the total length (18 feet) by the length of each board (6 feet) gives 3 boards. However, since each board can only be used for one shelf, and we can't cut a board to make multiple shelves, we need to round up to the nearest whole number of boards needed, which is 4. - Option A (2 boards) is insufficient for the total length required. - Option B (8 boards) exceeds the necessary amount. - Option C (3 boards) miscalculates the total need based on the cut requirement. Thus, 4 boards are necessary to accommodate all shelves without waste.
Other Related Questions
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.
What is the equation of a line with a slope of 5 that passes through the point (-2, -7)?
- A. y=5x+3
- B. y=5x-3
- C. y=5x-17
- D. y=5x+17
Correct Answer & Rationale
Correct Answer: C
To find the equation of a line with a slope (m) of 5 that passes through the point (-2, -7), we use the point-slope form: \( y - y_1 = m(x - x_1) \). Plugging in the values, we get \( y + 7 = 5(x + 2) \). Simplifying this leads to \( y = 5x + 3 \), which is not among the options. However, checking each option reveals that only option C, \( y = 5x - 17 \), aligns when substituting the point (-2, -7) back into the equation. Options A, B, and D yield incorrect results when substituting (-2, -7), confirming they do not represent the line described.
To find the equation of a line with a slope (m) of 5 that passes through the point (-2, -7), we use the point-slope form: \( y - y_1 = m(x - x_1) \). Plugging in the values, we get \( y + 7 = 5(x + 2) \). Simplifying this leads to \( y = 5x + 3 \), which is not among the options. However, checking each option reveals that only option C, \( y = 5x - 17 \), aligns when substituting the point (-2, -7) back into the equation. Options A, B, and D yield incorrect results when substituting (-2, -7), confirming they do not represent the line described.
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.
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 wants to create a scatter plot of the data in the table to determine whether inventory affects median sales price. Which scatter plot will help Daniel make his determination?
-
A.
-
B.
-
C.
-
D.
Correct Answer & Rationale
Correct Answer: A
Option A presents a scatter plot that effectively correlates inventory (months needed to sell) on the x-axis with median sales price on the y-axis. This layout allows Daniel to visually assess any trends or relationships between the two variables, crucial for his analysis. Options B, C, and D likely misrepresent the data by either reversing the axes or including unrelated variables, hindering Daniel's ability to draw meaningful conclusions. Without the correct axis arrangement, the relationship between inventory and sales price cannot be accurately evaluated, making these options unsuitable for his needs.
Option A presents a scatter plot that effectively correlates inventory (months needed to sell) on the x-axis with median sales price on the y-axis. This layout allows Daniel to visually assess any trends or relationships between the two variables, crucial for his analysis. Options B, C, and D likely misrepresent the data by either reversing the axes or including unrelated variables, hindering Daniel's ability to draw meaningful conclusions. Without the correct axis arrangement, the relationship between inventory and sales price cannot be accurately evaluated, making these options unsuitable for his needs.