The distance, d, in feet, it takes to come to a complete stop when driving a car r miles per hour can be found using the equation d = 1/20(r^2)+ r. If it takes a car 240 feet to come to a complete stop, what was the speed of the car, in miles per hour, when the driver began to stop it?
- A. 40
- B. 30
- C. 60
- D. 80
Correct Answer & Rationale
Correct Answer: A
To find the speed of the car when it takes 240 feet to stop, substitute d = 240 into the equation d = 1/20(r^2) + r. This leads to the equation 240 = 1/20(r^2) + r. Multiplying through by 20 simplifies to 4800 = r^2 + 20r, which rearranges to r^2 + 20r - 4800 = 0. Solving this quadratic equation yields r = 40 or r = -120. Since speed cannot be negative, the valid solution is 40 mph. Option B (30) does not satisfy the equation, leading to a shorter stopping distance. Option C (60) results in a stopping distance of 480 feet, which exceeds 240 feet. Option D (80) produces a stopping distance of 800 feet, also incorrect. Thus, only 40 mph meets the criteria.
To find the speed of the car when it takes 240 feet to stop, substitute d = 240 into the equation d = 1/20(r^2) + r. This leads to the equation 240 = 1/20(r^2) + r. Multiplying through by 20 simplifies to 4800 = r^2 + 20r, which rearranges to r^2 + 20r - 4800 = 0. Solving this quadratic equation yields r = 40 or r = -120. Since speed cannot be negative, the valid solution is 40 mph. Option B (30) does not satisfy the equation, leading to a shorter stopping distance. Option C (60) results in a stopping distance of 480 feet, which exceeds 240 feet. Option D (80) produces a stopping distance of 800 feet, also incorrect. Thus, only 40 mph meets the criteria.
Other Related Questions
A landscape worker is building a rock wall around a triangular flower garden. He has completed the rock wall on two sides of the garden.
The perimeter of the garden is 239 feet. What is the length, in feet, of the rock wall that the worker still needs to complete?
- A. 101
- B. 185
- C. 54
- D. 138
Correct Answer & Rationale
Correct Answer: D
To determine the length of the rock wall still needed, first, the total perimeter of the triangular garden is 239 feet. The worker has already completed two sides, leaving one side to be built. To find the length of the remaining side, we subtract the lengths of the two completed sides from the total perimeter. The answer of 138 feet indicates that the lengths of the two sides combined equal 101 feet (239 - 138 = 101). Option A (101) represents the combined length of the two completed sides, not the remaining side. Option B (185) exceeds the total perimeter, which is impossible. Option C (54) does not fit the calculations based on the perimeter. Thus, only option D accurately reflects the length of the remaining side to complete the wall.
To determine the length of the rock wall still needed, first, the total perimeter of the triangular garden is 239 feet. The worker has already completed two sides, leaving one side to be built. To find the length of the remaining side, we subtract the lengths of the two completed sides from the total perimeter. The answer of 138 feet indicates that the lengths of the two sides combined equal 101 feet (239 - 138 = 101). Option A (101) represents the combined length of the two completed sides, not the remaining side. Option B (185) exceeds the total perimeter, which is impossible. Option C (54) does not fit the calculations based on the perimeter. Thus, only option D accurately reflects the length of the remaining side to complete the wall.
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.
The graph shows a handyman's fees, f(x), in terms of the hours worked, x. The fees include a fuel charge and an hourly rate. What is the handyman's hourly rate?
- A. $5
- B. $55
- C. $30
- D. $25
Correct Answer & Rationale
Correct Answer: D
To determine the handyman's hourly rate, we analyze the graph showing the relationship between fees and hours worked. The hourly rate is represented by the slope of the line on the graph. Option A ($5) is too low for a reasonable hourly rate in this context. Option B ($55) is excessively high, suggesting an unrealistic fee for common handyman services. Option C ($30) may seem plausible, but it does not match the slope indicated by the graph. Option D ($25) accurately reflects the slope calculated from the graph, representing a fair and competitive hourly rate for handyman services.
To determine the handyman's hourly rate, we analyze the graph showing the relationship between fees and hours worked. The hourly rate is represented by the slope of the line on the graph. Option A ($5) is too low for a reasonable hourly rate in this context. Option B ($55) is excessively high, suggesting an unrealistic fee for common handyman services. Option C ($30) may seem plausible, but it does not match the slope indicated by the graph. Option D ($25) accurately reflects the slope calculated from the graph, representing a fair and competitive hourly rate for handyman services.
Fix It Fast is an auto repair shop that employs 10 mechanics. Each day, the shop owner randomly picks 1 mechanic to receive a free lunch. What is the probability the shop owner will pick the same mechanic to receive a free lunch 2 days in a row?
- A. 1\20
- B. 1/100
- C. 1\5
- D. 1\10
Correct Answer & Rationale
Correct Answer: B
To determine the probability of picking the same mechanic two days in a row, we start by recognizing that there are 10 mechanics. On the first day, any mechanic can be chosen, which does not affect the overall probability. On the second day, to pick the same mechanic again, there is only 1 favorable outcome (the chosen mechanic) out of 10 possible mechanics. Thus, the probability of selecting that same mechanic on the second day is 1/10. Since the first day's choice does not influence this, we multiply the probabilities: (1/10) * (1/10) = 1/100. - Option A (1/20) is incorrect as it miscalculates the favorable outcomes. - Option C (1/5) incorrectly assumes a higher likelihood without considering the second day's requirement. - Option D (1/10) only reflects the probability of picking a mechanic on day two, not the two-day scenario.
To determine the probability of picking the same mechanic two days in a row, we start by recognizing that there are 10 mechanics. On the first day, any mechanic can be chosen, which does not affect the overall probability. On the second day, to pick the same mechanic again, there is only 1 favorable outcome (the chosen mechanic) out of 10 possible mechanics. Thus, the probability of selecting that same mechanic on the second day is 1/10. Since the first day's choice does not influence this, we multiply the probabilities: (1/10) * (1/10) = 1/100. - Option A (1/20) is incorrect as it miscalculates the favorable outcomes. - Option C (1/5) incorrectly assumes a higher likelihood without considering the second day's requirement. - Option D (1/10) only reflects the probability of picking a mechanic on day two, not the two-day scenario.