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
Dr. Evers is experimenting with light beams and prisms. He passes a beam of white light through a triangular prism which spreads the light out into its six rainbow colors. The bases of the prism are equilateral triangles. The surface area of this prism is 4,292 square millimeters. The area of each triangular face is 271 square millimeters. Which expression can be used to find h, the height, in millimeters, of the prism?
- A. 4,292/3(25)
- B. 4,292/271
- C. (4,292-271)/25
- D. (4,292-2(271))/3(25)
Correct Answer & Rationale
Correct Answer: D
To find the height \( h \) of the prism, we start with the total surface area of the prism, which includes the two triangular bases and three rectangular sides. The area of the two triangular bases is \( 2 \times 271 = 542 \) square millimeters. Subtracting this from the total surface area gives \( 4,292 - 542 = 3,750 \) square millimeters for the area of the rectangular sides. Since the height \( h \) is involved in the area of the rectangles, dividing this area by the perimeter of the base (which is \( 3 \times 25 = 75 \) mm) leads to \( h = \frac{3,750}{75} \) or \( \frac{4,292 - 542}{75} \), simplifying to option D. Options A and B incorrectly compute the height without accounting for the rectangular areas properly. Option C miscalculates the area of the triangular bases and does not consider the full surface area needed to find \( h \). Thus, only option D correctly utilizes the total surface area and the dimensions of the prism to derive the height.
To find the height \( h \) of the prism, we start with the total surface area of the prism, which includes the two triangular bases and three rectangular sides. The area of the two triangular bases is \( 2 \times 271 = 542 \) square millimeters. Subtracting this from the total surface area gives \( 4,292 - 542 = 3,750 \) square millimeters for the area of the rectangular sides. Since the height \( h \) is involved in the area of the rectangles, dividing this area by the perimeter of the base (which is \( 3 \times 25 = 75 \) mm) leads to \( h = \frac{3,750}{75} \) or \( \frac{4,292 - 542}{75} \), simplifying to option D. Options A and B incorrectly compute the height without accounting for the rectangular areas properly. Option C miscalculates the area of the triangular bases and does not consider the full surface area needed to find \( h \). Thus, only option D correctly utilizes the total surface area and the dimensions of the prism to derive the height.
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.
The owner of a small cookie shop is examining the shop's revenue and costs to see how she can increase profits. Currently, the shop has expenses of $41.26 and $0.19 per cookie.
The shop's revenue and profit depend on the sales price of the cookies. The daily revenue is given in the graph below, where x is the sales price of the cookies and y is the expected revenue at that price.
The owner has decided to take out a loan to purchase updated equipment. A bank has agreed to loan the owner $2,000 for the purchase of the equipment at a simple interest rate of 4.69% payable annually.
To the nearest dollar, what is the total amount the shop owner will pay on the loan over the 3 years?
- A. $2,028
- B. $2,276
- C. $2,760
- D. $2,092
Correct Answer & Rationale
Correct Answer: B
To calculate the total amount paid on a loan, the formula for simple interest is used: Total Payment = Principal + (Principal × Rate × Time). For a $2,000 loan at 4.69% interest over 3 years, the interest is calculated as $2,000 × 0.0469 × 3 = $280.40. Adding this to the principal gives a total of $2,280.40, which rounds to $2,276. Option A ($2,028) underestimates the interest accrued. Option C ($2,760) incorrectly assumes a higher interest rate or longer term. Option D ($2,092) miscalculates the total by not accounting for the full interest over 3 years.
To calculate the total amount paid on a loan, the formula for simple interest is used: Total Payment = Principal + (Principal × Rate × Time). For a $2,000 loan at 4.69% interest over 3 years, the interest is calculated as $2,000 × 0.0469 × 3 = $280.40. Adding this to the principal gives a total of $2,280.40, which rounds to $2,276. Option A ($2,028) underestimates the interest accrued. Option C ($2,760) incorrectly assumes a higher interest rate or longer term. Option D ($2,092) miscalculates the total by not accounting for the full interest over 3 years.
Last weekend, 625 runners entered a 10,000-meter race. A 10,000- meter race is 6.2 miles long. Ruben won the race with a finishing time of 29 minutes 51 seconds.
The graphs show information about the top 10 runners.
Based on the histogram, which statement describes the finishing time of the runner in position 3?
- A. The finishing time was between 30 and 31 minutes
- B. The finishing time was between 33 and 34 minutes
- C. The finishing time was between 31 and 32 minutes
- D. The finishing time was between 32 and 33 minutes
Correct Answer & Rationale
Correct Answer: C
The finishing time for the runner in position 3 falls within the 31 to 32 minutes range, as indicated by the histogram. This range is supported by the data distribution, showing that the majority of runners in the top 10 finished within this timeframe. Option A is incorrect because it suggests a time between 30 and 31 minutes, which does not align with the position 3 runner's time. Option B is inaccurate, as it indicates a finishing time between 33 and 34 minutes, which is too high for this position. Option D, while close, incorrectly suggests a time between 32 and 33 minutes, which does not match the histogram data for the third place runner.
The finishing time for the runner in position 3 falls within the 31 to 32 minutes range, as indicated by the histogram. This range is supported by the data distribution, showing that the majority of runners in the top 10 finished within this timeframe. Option A is incorrect because it suggests a time between 30 and 31 minutes, which does not align with the position 3 runner's time. Option B is inaccurate, as it indicates a finishing time between 33 and 34 minutes, which is too high for this position. Option D, while close, incorrectly suggests a time between 32 and 33 minutes, which does not match the histogram data for the third place runner.