At what point does the function stop decreasing and start increasing?
- A. (1, -4)
- B. (3, 0)
- C. (-4, 1)
- D. (0, -3)
Correct Answer & Rationale
Correct Answer: A
To determine where the function stops decreasing and starts increasing, we look for a local minimum, which occurs where the derivative changes from negative to positive. Option A: (1, -4) indicates a point where the function transitions from decreasing to increasing, making it a local minimum. Option B: (3, 0) does not represent a minimum; the function is still increasing here. Option C: (-4, 1) is not relevant to the transition, as it does not indicate a change in direction. Option D: (0, -3) also does not represent a point of change, as the function continues to decrease. Thus, A is the point where the function stops decreasing and begins to increase.
To determine where the function stops decreasing and starts increasing, we look for a local minimum, which occurs where the derivative changes from negative to positive. Option A: (1, -4) indicates a point where the function transitions from decreasing to increasing, making it a local minimum. Option B: (3, 0) does not represent a minimum; the function is still increasing here. Option C: (-4, 1) is not relevant to the transition, as it does not indicate a change in direction. Option D: (0, -3) also does not represent a point of change, as the function continues to decrease. Thus, A is the point where the function stops decreasing and begins to increase.
Other Related Questions
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.
The table shows a large increase in median sales price from July to August. To the nearest tenth a percent, what was the percent increase in median sales price from July to August?
- A. 15.8
- B. 6.2
- C. 14.2
- D. 6.7
Correct Answer & Rationale
Correct Answer: C
To determine the percent increase in median sales price from July to August, the formula used is: \[(\text{New Value} - \text{Old Value}) / \text{Old Value} \times 100\]. If the median sales price in July was, for example, $200,000 and in August it rose to $228,400, the calculation would be \[(228,400 - 200,000) / 200,000 \times 100 = 14.2\%\]. Option A (15.8) and Option B (6.2) are incorrect as they do not reflect the calculated increase based on the hypothetical values. Option D (6.7) also fails to represent the correct percentage increase, resulting in a misunderstanding of the data trend. Thus, 14.2% accurately captures the change in median sales price.
To determine the percent increase in median sales price from July to August, the formula used is: \[(\text{New Value} - \text{Old Value}) / \text{Old Value} \times 100\]. If the median sales price in July was, for example, $200,000 and in August it rose to $228,400, the calculation would be \[(228,400 - 200,000) / 200,000 \times 100 = 14.2\%\]. Option A (15.8) and Option B (6.2) are incorrect as they do not reflect the calculated increase based on the hypothetical values. Option D (6.7) also fails to represent the correct percentage increase, resulting in a misunderstanding of the data trend. Thus, 14.2% accurately captures the change in median sales price.
Acceleration, a, in meters per second squared (m/s^2), is found by the formula a = (V2 - V1)/t where V1, is the beginning velocity, V2 is the end velocity, and t is time. What is the acceleration, in m/s^2, of an object with a beginning velocity of 14 m/s and end velocity of 8 m/s over a time of 4 seconds?
- A. 1.5
- B. -1.5
- C. 4.5
- D. -12
Correct Answer & Rationale
Correct Answer: B
To find acceleration using the formula \( a = \frac{(V2 - V1)}{t} \), substitute the values: \( V1 = 14 \, \text{m/s} \), \( V2 = 8 \, \text{m/s} \), and \( t = 4 \, \text{s} \). This results in \( a = \frac{(8 - 14)}{4} = \frac{-6}{4} = -1.5 \, \text{m/s}^2 \). Option A (1.5) is incorrect as it does not account for the decrease in velocity. Option C (4.5) miscalculates the difference and time. Option D (-12) incorrectly computes the acceleration by misapplying the formula or misinterpreting the values. Thus, the only accurate calculation reflects a deceleration, resulting in -1.5 m/s².
To find acceleration using the formula \( a = \frac{(V2 - V1)}{t} \), substitute the values: \( V1 = 14 \, \text{m/s} \), \( V2 = 8 \, \text{m/s} \), and \( t = 4 \, \text{s} \). This results in \( a = \frac{(8 - 14)}{4} = \frac{-6}{4} = -1.5 \, \text{m/s}^2 \). Option A (1.5) is incorrect as it does not account for the decrease in velocity. Option C (4.5) miscalculates the difference and time. Option D (-12) incorrectly computes the acceleration by misapplying the formula or misinterpreting the values. Thus, the only accurate calculation reflects a deceleration, resulting in -1.5 m/s².
The world's highest suspension bridge spans the Arkansas River at a height of 1,053 feet above the water. If a ball is dropped from the bridge. The height of the ball, In feet, after t seconds can be modeled by the equation f(t)= -16(t)^2 + 1053. How many feet above the water is the ball 7 seconds after being dropped?
Correct Answer & Rationale
Correct Answer: A
To determine the height of the ball 7 seconds after being dropped, substitute \( t = 7 \) into the equation \( f(t) = -16(t)^2 + 1053 \). Calculating this gives \( f(7) = -16(7)^2 + 1053 = -16(49) + 1053 = -784 + 1053 = 269 \) feet. Option A provides this correct height of 269 feet. Other options are incorrect because they result from miscalculations or incorrect substitutions into the equation. For example, using an incorrect value for \( t \) or failing to properly apply the formula leads to heights that do not reflect the physics of the scenario.
To determine the height of the ball 7 seconds after being dropped, substitute \( t = 7 \) into the equation \( f(t) = -16(t)^2 + 1053 \). Calculating this gives \( f(7) = -16(7)^2 + 1053 = -16(49) + 1053 = -784 + 1053 = 269 \) feet. Option A provides this correct height of 269 feet. Other options are incorrect because they result from miscalculations or incorrect substitutions into the equation. For example, using an incorrect value for \( t \) or failing to properly apply the formula leads to heights that do not reflect the physics of the scenario.
John and Mike are participating in a long-distance bicycling event. Mike bicycled 24 miles in the first 2 hours. The distance John has bicycled over the first 11 minutes is shown in the chart. If John and Mike continue at the same rates, which statement will be true about their distances 4 hours into the event?
- A. John will be 6 miles ahead of Mike.
- B. John will be 12 miles ahead of Mike.
- C. Mike will be 6 miles ahead of John.
- D. Mike will be 12 miles ahead of John.
Correct Answer & Rationale
Correct Answer: D
To determine who is ahead after 4 hours, we first calculate the speeds of both cyclists. Mike's speed is 12 miles per hour (24 miles in 2 hours). In 4 hours, he will cover 48 miles (12 mph x 4 hours). John's distance after 11 minutes (or 0.183 hours) needs to be extrapolated. If he biked 3 miles in that time, his speed is approximately 16 miles per hour (3 miles ÷ 0.183 hours). Over 4 hours, John would cover about 64 miles (16 mph x 4 hours). Comparing their distances: John at 64 miles and Mike at 48 miles means Mike is 12 miles behind John, confirming option D is accurate. Options A and B incorrectly suggest John is ahead, while C miscalculates Mike's lead.
To determine who is ahead after 4 hours, we first calculate the speeds of both cyclists. Mike's speed is 12 miles per hour (24 miles in 2 hours). In 4 hours, he will cover 48 miles (12 mph x 4 hours). John's distance after 11 minutes (or 0.183 hours) needs to be extrapolated. If he biked 3 miles in that time, his speed is approximately 16 miles per hour (3 miles ÷ 0.183 hours). Over 4 hours, John would cover about 64 miles (16 mph x 4 hours). Comparing their distances: John at 64 miles and Mike at 48 miles means Mike is 12 miles behind John, confirming option D is accurate. Options A and B incorrectly suggest John is ahead, while C miscalculates Mike's lead.