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
The weight of a red blood cell is about 4.5 × 10*11 grams. A blood sample has 1.6 × 10 red blood cells. What is the total weight, in grams, of red blood cells in the sample the answer with the correct scientific notation.
- A. 2.9 × 10^18
- B. 7.2 × 10^(-4)
- C. 7.2 × 10^(-77)
- D. 6.1 × 10^(-4)
Correct Answer & Rationale
Correct Answer: B
To find the total weight of the red blood cells, multiply the weight of one red blood cell (4.5 × 10^-11 grams) by the total number of cells (1.6 × 10^6). This calculation yields 7.2 × 10^-5 grams, which can be expressed in scientific notation as 7.2 × 10^(-4) grams. Option A (2.9 × 10^18) is incorrect because it suggests an unrealistically high total weight, indicating a misunderstanding of scientific notation. Options C (7.2 × 10^(-77)) and D (6.1 × 10^(-4)) also fail to represent the correct multiplication, with C being far too small and D lacking accuracy in the calculated value.
To find the total weight of the red blood cells, multiply the weight of one red blood cell (4.5 × 10^-11 grams) by the total number of cells (1.6 × 10^6). This calculation yields 7.2 × 10^-5 grams, which can be expressed in scientific notation as 7.2 × 10^(-4) grams. Option A (2.9 × 10^18) is incorrect because it suggests an unrealistically high total weight, indicating a misunderstanding of scientific notation. Options C (7.2 × 10^(-77)) and D (6.1 × 10^(-4)) also fail to represent the correct multiplication, with C being far too small and D lacking accuracy in the calculated value.
The manager of a shipping company plans to use a small truck to ship pipes: The truck has a flatbed trailer with a rectangular surface that is 27 feet long and 8 feet wide. The truck will travel from Atherton to Bakersfield, where some pipes will be delivered, and then on to Castlewood to deliver the remaining pipes. The map shows the roads that connect Atherton. Bakersfield. and Castlewood.
The manager is planning to buy a new truck with better gas mileage. He collected data bout the gas mileage of one of the company's trucks. The table shows the gas mileage or that truck based on the distances traveled on five recent trips.
How many different ways can the truck travel from Atherton to Bakersfield a to Castlewood, using the roads on the map?
- A. 6
- B. 8
- C. 9
- D. 5
Correct Answer & Rationale
Correct Answer: A
To determine the number of different routes from Atherton to Bakersfield and then to Castlewood, we analyze the connections between these locations. If there are 3 distinct paths from Atherton to Bakersfield and 2 distinct paths from Bakersfield to Castlewood, the total number of combinations is found by multiplying the number of options: 3 paths (Atherton to Bakersfield) × 2 paths (Bakersfield to Castlewood) = 6 routes. Options B (8), C (9), and D (5) miscalculate the available paths or overlook the combinations of routes, leading to incorrect totals. Thus, the correct answer accurately reflects the possible travel routes.
To determine the number of different routes from Atherton to Bakersfield and then to Castlewood, we analyze the connections between these locations. If there are 3 distinct paths from Atherton to Bakersfield and 2 distinct paths from Bakersfield to Castlewood, the total number of combinations is found by multiplying the number of options: 3 paths (Atherton to Bakersfield) × 2 paths (Bakersfield to Castlewood) = 6 routes. Options B (8), C (9), and D (5) miscalculate the available paths or overlook the combinations of routes, leading to incorrect totals. Thus, the correct answer accurately reflects the possible travel routes.
On Monday; Alicia buys x shirts at $8 each and y slacks at $25 each. On Wednesday, Alicia returns 2 pairs of slacks. Which expression represents the total value of her purchases?
- A. 8x + 23y
- B. 8x + 25(y - 2)
- C. 8x - 2) + 25y
- D. 8x + 25y - 2
Correct Answer & Rationale
Correct Answer: B
To calculate the total value of Alicia's purchases, we need to account for the cost of shirts and slacks, as well as the return of 2 pairs of slacks. Option B, \(8x + 25(y - 2)\), correctly reflects the initial cost of \(x\) shirts at $8 each and \(y\) slacks at $25 each, while subtracting the cost of the 2 returned slacks, which is \(2 \times 25\). Option A, \(8x + 23y\), incorrectly reduces the price of slacks to $23, which is not stated in the problem. Option C, \(8x - 2 + 25y\), miscalculates by subtracting $2 instead of the cost of the returned slacks. Option D, \(8x + 25y - 2\), also incorrectly subtracts $2 instead of the total cost of the slacks returned.
To calculate the total value of Alicia's purchases, we need to account for the cost of shirts and slacks, as well as the return of 2 pairs of slacks. Option B, \(8x + 25(y - 2)\), correctly reflects the initial cost of \(x\) shirts at $8 each and \(y\) slacks at $25 each, while subtracting the cost of the 2 returned slacks, which is \(2 \times 25\). Option A, \(8x + 23y\), incorrectly reduces the price of slacks to $23, which is not stated in the problem. Option C, \(8x - 2 + 25y\), miscalculates by subtracting $2 instead of the cost of the returned slacks. Option D, \(8x + 25y - 2\), also incorrectly subtracts $2 instead of the total cost of the slacks returned.
Which pair of equations represents parallel lines?
- A. -2x + y + 2 = 0, y = -(1/2)x - 4
- B. 3x + y = -8, y = 3x - 8
- C. x + 2y = 8, -x - 2y = 3
- D. -(2/3)x + y = 12, y = -(3/2)x - 1
Correct Answer & Rationale
Correct Answer: C
To identify parallel lines, the slopes of the equations must be equal. Option A has slopes of 1/2 and -1/2, which are not equal. Option B has slopes of 3 and 3, indicating the lines are parallel; however, it is not the correct answer as it does not match the requirement for both equations. Option C has the first equation rearranged to slope -1/2 and the second to slope -1/2, confirming they are parallel. Option D features slopes of 2/3 and -3/2, which are also not equal, indicating the lines intersect. Thus, only option C accurately represents parallel lines.
To identify parallel lines, the slopes of the equations must be equal. Option A has slopes of 1/2 and -1/2, which are not equal. Option B has slopes of 3 and 3, indicating the lines are parallel; however, it is not the correct answer as it does not match the requirement for both equations. Option C has the first equation rearranged to slope -1/2 and the second to slope -1/2, confirming they are parallel. Option D features slopes of 2/3 and -3/2, which are also not equal, indicating the lines intersect. Thus, only option C accurately represents parallel lines.