A diver jumps from a platform. The height, h meters, the diver is above the water t seconds after jumping is represented by h = -16t^2 + 16t + 6.5. To the near hundredth of a second, how many seconds after jumping is the diver 2.5 meters above the water?
- A. 2.79
- B. 1.32
- C. 2.83
- D. 1.21
Correct Answer & Rationale
Correct Answer: D
To find when the diver is 2.5 meters above the water, substitute h = 2.5 into the equation: \[ 2.5 = -16t^2 + 16t + 6.5. \] Rearranging gives: \[ -16t^2 + 16t + 4 = 0. \] Using the quadratic formula, we solve for t, yielding two potential solutions. The option D (1.21 seconds) is valid as it falls within the realistic time frame of the jump. Options A (2.79) and C (2.83) exceed the expected time of descent, while B (1.32) does not satisfy the equation, confirming that only D accurately represents the diver's position at 2.5 meters above the water.
To find when the diver is 2.5 meters above the water, substitute h = 2.5 into the equation: \[ 2.5 = -16t^2 + 16t + 6.5. \] Rearranging gives: \[ -16t^2 + 16t + 4 = 0. \] Using the quadratic formula, we solve for t, yielding two potential solutions. The option D (1.21 seconds) is valid as it falls within the realistic time frame of the jump. Options A (2.79) and C (2.83) exceed the expected time of descent, while B (1.32) does not satisfy the equation, confirming that only D accurately represents the diver's position at 2.5 meters above the water.
Other Related Questions
For an emergency service call, a plumbing company charges a flat fee of $60 plus $40 an hour. A customer remembers paying at least $200 for an emergency service. Which phrase describes the number of hours the plumbing company was at the service call?
- A. at most 2 hours
- B. at most 3.5 hours
- C. at least 3.5 hours
- D. at least 2 hours
Correct Answer & Rationale
Correct Answer: C
To determine the number of hours the plumbing company was on the service call, we start with the total charge of at least $200. The charge consists of a flat fee of $60 plus $40 per hour. First, subtract the flat fee from the total: $200 - $60 = $140. Next, divide this by the hourly rate: $140 รท $40 = 3.5 hours. This indicates that the service lasted at least 3.5 hours. Option A (at most 2 hours) is incorrect, as 2 hours would only cost $140. Option B (at most 3.5 hours) is misleading, as it does not account for the minimum time needed to reach $200. Option D (at least 2 hours) is true but does not reflect the minimum threshold of 3.5 hours. Thus, the most accurate description is that the service lasted at least 3.5 hours.
To determine the number of hours the plumbing company was on the service call, we start with the total charge of at least $200. The charge consists of a flat fee of $60 plus $40 per hour. First, subtract the flat fee from the total: $200 - $60 = $140. Next, divide this by the hourly rate: $140 รท $40 = 3.5 hours. This indicates that the service lasted at least 3.5 hours. Option A (at most 2 hours) is incorrect, as 2 hours would only cost $140. Option B (at most 3.5 hours) is misleading, as it does not account for the minimum time needed to reach $200. Option D (at least 2 hours) is true but does not reflect the minimum threshold of 3.5 hours. Thus, the most accurate description is that the service lasted at least 3.5 hours.
Which table shows a function?
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A.
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B.
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C.
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D.
Correct Answer & Rationale
Correct Answer: A
To determine which table represents a function, we look for a unique output for every input. Option A demonstrates this principle, as each input corresponds to a single output, confirming a functional relationship. In contrast, Option B features repeated inputs yielding different outputs, violating the definition of a function. Option C also presents multiple outputs for the same input, disqualifying it as a function. Lastly, Option D has inputs linked to multiple outputs as well, further indicating it does not represent a function. Thus, only Option A adheres to the criteria for a function.
To determine which table represents a function, we look for a unique output for every input. Option A demonstrates this principle, as each input corresponds to a single output, confirming a functional relationship. In contrast, Option B features repeated inputs yielding different outputs, violating the definition of a function. Option C also presents multiple outputs for the same input, disqualifying it as a function. Lastly, Option D has inputs linked to multiple outputs as well, further indicating it does not represent a function. Thus, only Option A adheres to the criteria for a function.
The equation d/f = g represents gallons of gasoline used, g, in terms of distance traveled in miles, d, and fuel efficiency, / miles per gallon of gasoline. Which combination of distance traveled and fuel efficiency uses 3 gallons of gasoline?
- A. 7 miles and 21 miles per gallon
- B. 57 miles and 19 miles per gallon
- C. 23 miles and 20 miles per gallon
- D. 32 miles and 35 miles per gallon
Correct Answer & Rationale
Correct Answer: B
To determine which combination uses 3 gallons of gasoline, we can rearrange the equation d/f = g to find d = g * f. For g = 3 gallons, we calculate d for each option. A: 7 miles and 21 mpg results in d = 3 * 21 = 63 miles, which is incorrect. B: 57 miles and 19 mpg gives d = 3 * 19 = 57 miles, matching the distance traveled. C: 23 miles and 20 mpg leads to d = 3 * 20 = 60 miles, which is incorrect. D: 32 miles and 35 mpg results in d = 3 * 35 = 105 miles, which is also incorrect. Only option B correctly satisfies the equation for 3 gallons of gasoline used.
To determine which combination uses 3 gallons of gasoline, we can rearrange the equation d/f = g to find d = g * f. For g = 3 gallons, we calculate d for each option. A: 7 miles and 21 mpg results in d = 3 * 21 = 63 miles, which is incorrect. B: 57 miles and 19 mpg gives d = 3 * 19 = 57 miles, matching the distance traveled. C: 23 miles and 20 mpg leads to d = 3 * 20 = 60 miles, which is incorrect. D: 32 miles and 35 mpg results in d = 3 * 35 = 105 miles, which is also incorrect. Only option B correctly satisfies the equation for 3 gallons of gasoline used.
What is the slope of a line perpendicular to the line given by the equation 5x - 2y = -10?
- A. -0.4
- B. 2\5
- C. 5\2
- D. -2.5
Correct Answer & Rationale
Correct Answer: B
To find the slope of a line perpendicular to the given equation \(5x - 2y = -10\), we first convert it to slope-intercept form (y = mx + b). Rearranging gives \(y = \frac{5}{2}x + 5\), revealing a slope (m) of \(\frac{5}{2}\). The slope of a line perpendicular to another is the negative reciprocal, which is \(-\frac{2}{5}\). Option A (-0.4) is equivalent to \(-\frac{2}{5}\), which is incorrect as it represents a decimal form. Option C (\(\frac{5}{2}\)) is the slope of the original line, not its perpendicular. Option D (-2.5) does not represent the correct negative reciprocal either.
To find the slope of a line perpendicular to the given equation \(5x - 2y = -10\), we first convert it to slope-intercept form (y = mx + b). Rearranging gives \(y = \frac{5}{2}x + 5\), revealing a slope (m) of \(\frac{5}{2}\). The slope of a line perpendicular to another is the negative reciprocal, which is \(-\frac{2}{5}\). Option A (-0.4) is equivalent to \(-\frac{2}{5}\), which is incorrect as it represents a decimal form. Option C (\(\frac{5}{2}\)) is the slope of the original line, not its perpendicular. Option D (-2.5) does not represent the correct negative reciprocal either.