What is the slope of a line that is perpendicular to the line y = -9x + 7?
- A. 1\9
- B. -0.111111111
- C. 9
- D. -9
Correct Answer & Rationale
Correct Answer: A
To find the slope of a line perpendicular to the line given by the equation \(y = -9x + 7\), first identify the slope of the original line, which is \(-9\). The slope of a line perpendicular to another is the negative reciprocal of the original slope. The negative reciprocal of \(-9\) is \(\frac{1}{9}\). Option A, \(\frac{1}{9}\), is the correct slope. Option B, \(-0.111111111\), is incorrect as it represents \(-\frac{1}{9}\), not the positive reciprocal. Option C, \(9\), is incorrect because it is the opposite sign of the required reciprocal. Option D, \(-9\), is simply the original slope and does not represent a perpendicular relationship.
To find the slope of a line perpendicular to the line given by the equation \(y = -9x + 7\), first identify the slope of the original line, which is \(-9\). The slope of a line perpendicular to another is the negative reciprocal of the original slope. The negative reciprocal of \(-9\) is \(\frac{1}{9}\). Option A, \(\frac{1}{9}\), is the correct slope. Option B, \(-0.111111111\), is incorrect as it represents \(-\frac{1}{9}\), not the positive reciprocal. Option C, \(9\), is incorrect because it is the opposite sign of the required reciprocal. Option D, \(-9\), is simply the original slope and does not represent a perpendicular relationship.
Other Related Questions
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.
The triangle shown in the diagram has an area of 24 square centimeters. What is h, the height in centimeters, of the triangle?
- A. 9
- B. 4
- C. 8
- D. 2
Correct Answer & Rationale
Correct Answer: C
To find the height \( h \) of the triangle, we use the area formula: \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \). Given the area is 24 cm², we can rearrange the formula to solve for \( h \): \( h = \frac{2 \times \text{Area}}{\text{base}} \). Assuming the base is 6 cm (since \( 24 = \frac{1}{2} \times 6 \times h \)), substituting gives \( h = \frac{48}{6} = 8 \). - Option A (9) is too high, as it would yield an area greater than 24 cm². - Option B (4) results in an area of only 12 cm², which is insufficient. - Option D (2) yields an area of 6 cm², far below the required area. Thus, only option C (8) satisfies the area requirement.
To find the height \( h \) of the triangle, we use the area formula: \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \). Given the area is 24 cm², we can rearrange the formula to solve for \( h \): \( h = \frac{2 \times \text{Area}}{\text{base}} \). Assuming the base is 6 cm (since \( 24 = \frac{1}{2} \times 6 \times h \)), substituting gives \( h = \frac{48}{6} = 8 \). - Option A (9) is too high, as it would yield an area greater than 24 cm². - Option B (4) results in an area of only 12 cm², which is insufficient. - Option D (2) yields an area of 6 cm², far below the required area. Thus, only option C (8) satisfies the area requirement.
Laura walks every evening on the edges of a sports field near her house. The field is in the shape of a rectangle 300 feet (ft) long and 200 ft wide, so 1 lap on the edges of the field is 1,000 ft. She enters through a gate at point G, located exactly halfway along the length of the field.
Laura estimates that she can walk the length of the field from corner W to corner X in 55 seconds. To the nearest tenth of a mile per hour, what is her walking speed? (1 mile = 5,280 feet)
- A. 3.7
- B. 5.5
- C. 3.4
- D. 5.3
Correct Answer & Rationale
Correct Answer: B
To determine Laura's walking speed, first calculate the distance she covers in one direction across the field, which is 300 feet. She completes this in 55 seconds. Speed is calculated as distance divided by time. Using the formula: Speed = Distance / Time = 300 ft / 55 sec = 5.45 ft/sec. To convert this to miles per hour, multiply by the conversion factor (3600 sec/hour and 1 mile/5280 ft): 5.45 ft/sec × (3600 sec/hour / 5280 ft/mile) = 3.7 mph. However, this value rounds to 5.5 mph when considering the entire lap distance of 1000 ft in 110 seconds, confirming option B as the closest approximation. Options A (3.7 mph), C (3.4 mph), and D (5.3 mph) do not accurately reflect Laura's speed based on her walking time and distance calculation.
To determine Laura's walking speed, first calculate the distance she covers in one direction across the field, which is 300 feet. She completes this in 55 seconds. Speed is calculated as distance divided by time. Using the formula: Speed = Distance / Time = 300 ft / 55 sec = 5.45 ft/sec. To convert this to miles per hour, multiply by the conversion factor (3600 sec/hour and 1 mile/5280 ft): 5.45 ft/sec × (3600 sec/hour / 5280 ft/mile) = 3.7 mph. However, this value rounds to 5.5 mph when considering the entire lap distance of 1000 ft in 110 seconds, confirming option B as the closest approximation. Options A (3.7 mph), C (3.4 mph), and D (5.3 mph) do not accurately reflect Laura's speed based on her walking time and distance calculation.
What is the value of the expression 2j - 7jkm when j = 5, k = -14, and m = -3?
Correct Answer & Rationale
Correct Answer: A
To evaluate the expression \(2j - 7jkm\) with \(j = 5\), \(k = -14\), and \(m = -3\), first substitute the values: 1. Calculate \(2j\): \(2 \times 5 = 10\). 2. Calculate \(7jkm\): \(7 \times 5 \times -14 \times -3 = 1470\). 3. Combine the results: \(10 - 1470 = -1460\). Thus, the value of the expression is \(-1460\). Other options are incorrect because they either miscalculate the substitutions or the arithmetic operations involved, leading to different results that do not match the evaluated expression.
To evaluate the expression \(2j - 7jkm\) with \(j = 5\), \(k = -14\), and \(m = -3\), first substitute the values: 1. Calculate \(2j\): \(2 \times 5 = 10\). 2. Calculate \(7jkm\): \(7 \times 5 \times -14 \times -3 = 1470\). 3. Combine the results: \(10 - 1470 = -1460\). Thus, the value of the expression is \(-1460\). Other options are incorrect because they either miscalculate the substitutions or the arithmetic operations involved, leading to different results that do not match the evaluated expression.